IMAP Web-Based Beliefs Survey Manual

IMAP Web-Based Beliefs SurveyManual

Table of Contents

Note that the bookmarks for this manual can be used to facilitate movement within the manual. For example, within each segment rubric, each element of that rubricis bookmarked separately. And, further, the bookmarks are nested; within some elements (e.g., Examples and Training-Exercise sets) the items are bookmarkedindividually. Each bookmark that has an arrow (on Macs) or a + (on PCs) beside it can be opened to show additional bookmarks within that section.

I. Welcome (Introduction, Contact Information, and System Requirements for the Survey)II. Brief History of DevelopmentIII. Coding Data

1. Introduction2. Beliefs Statements3. Coding Navigator (Linked to each Belief Rubric and to each Segment Rubric)4. Beliefs-Rubric Development: Rubrics of Rubrics5. Belief Rubrics: Rubric of Rubrics and IMAP Results for Each of Seven Beliefs6. Segment Rubrics: Description, Survey Item, Rubric Scores, Scoring Summary, Examples, Training Exercises and

Solutions, and IMAP Results for Each of 17 RubricsIV. Minimum Software RequirementsV. InstallationVI. Survey Functions

1. Customizing Survey2. Browse Survey3. Accessing Data4. Password Maintenance5. Restarting an Incomplete Survey6. Video Streaming

VII. Browse the IMAP Web-Based Beliefs Survey1. Browsing-the-Survey Introduction2. Table of Contents for Beliefs Survey3. IMAP Web-Based Beliefs Survey

VIII. About IMAP

IWelcome

Integrating Mathematics and Pedagogy (IMAP) Web-Based Beliefs Survey

Development of this Beliefs Survey was supported by a grant awarded by the Interagency Educational ResearchInitiative, National Science Foundation (NSF) (REC-9979902) to Randolph Philipp and Judith Sowder at San DiegoState University. The views expressed are those of the authors and do not necessarily reflect the views of NSF.

Introduction

This beliefs survey was designed to assess respondents’ beliefs about mathematics, beliefs about learning or

knowing mathematics, and beliefs about children’s learning and doing mathematics. Our target audience was

prospective and practicing elementary school teachers, although the survey has been used with others. This survey is

not designed to explicitly assess respondents’ beliefs about teaching mathematics, although inferences about these

beliefs can reasonably be made. We set out to construct a beliefs-assessment instrument that would go beyond the

typical approach of using Likert scales because we believed that beliefs are best assessed in context. Our instrument

is web-based, making possible use of video clips and minimal branching based on how one responds to particular

questions. Survey administrators can customize the beliefs survey, and responses are saved as Excel files to simplify

data collection and subsequent coding.

What Makes This Survey Special?

We took seriously the notion that an important means for assessing beliefs is to determine how respondents

interpret, react to, or profess to act within well-defined contexts. Unlike traditional Likert-scale instruments that provide

statements to which respondents agree to one degree or another, our survey requires respondents to use their own

words to interpret situations, predict results of particular actions, and make recommendations. We then make

inferences on the basis of their responses. Each belief is assessed by two or three rubrics associated with different

segments of the instrument. This instrument can be used to collect data from many people in a relatively brief period of

time, and the customizing feature will enable researchers to assess specific beliefs or select rubrics of interest in their

work.

For qualitative researchers, we envision this beliefs survey as another tool to enhance, not replace, interview and

observational data.

Housing the Beliefs Survey

This instrument has been designed to be loaded onto a web server either at a researcher’s site or on a

commercial server. Directions to those who oversee the running of the server are available.

Credits

This instrument is the result of many people’s efforts. Our research team first set out to determine the beliefs

we wanted to address, and we settled upon seven beliefs in three categories. Randy Philipp and Bonnie Schappelle

used these beliefs to develop the first draft of the beliefs instrument and directed the development of the instrument for

use on the World Wide Web. Rebecca Ambrose, Lisa Clement, and Jennifer Chauvot then undertook the immense

task of developing the rubrics for scoring the beliefs instrument, an undertaking that led to additional revisions to the

draft. Rebecca Ambrose assembled a group of visiting researchers to code our large-scale-study data; she and Lisa

Clement trained these coders and led the groups coding the data, leading to additional revisions to the instrument.

Final refinement of the beliefs instrument and organization of this manual were undertaken by Jason Smith, under

direction of Rebecca Ambrose. Bonnie Schappelle oversaw the final production of this manual. Randolph Philipp and

Judith Sowder served as the Principal Investigators of the grant that funded the work.

Viewing the Beliefs-Survey Video Clips

Links to Video Clips Within the Manual

Survey Items 7 and 9 in the IMAP Web-Based Beliefs Survey include video clips to which those completing the survey are asked torespond. In this manual, links to those items appear in two locations:1. Within the relevant segment rubrics (B3-S9, B4-S9, B5-S7, B6-S9, and B7-S7) under Survey Item (e.g., Survey Item for B3-S9 ) .2. Within the Browse version of the survey for all segments of Items 7 and 9 (e.g., Items 7.1 )

After you watch the video clip, Do Not click the Back link on the video page to return to the manual page from which you accessed thevideo. The Back link on the video page will take you within the survey instead of back to the manual.

Instead, to return to the manual page from which you linked to the video clip,

Mac Users a) if you have downloaded the manual, use the Acrobat back arrow at the bottom of the page. b) if you are viewing the manual online, use the Internet Explorer or Netscape Back button.

PC Users a) if you have downloaded the manual, use the Acrobat back arrow at the bottom of the page. b) if you are viewing the manual online, the Back button for Internet Explorer or Netscape will not return you to the page from

which you linked to the video. Instead, the Back button takes you to the beginning of the manual. To avoid this problem, at http://www.sci.sdsu.edu/CRMSE/IMAP/pubs.html, choose the Download Manual link instead of theView Manual link.

Contact Information

The IMAP Web-Based Beliefs Survey is available on CD. To obtain the CD, please contactIMAP6475 Alvarado Road Suite 206San Diego, CA 92120

Fax (619) 594-1581E-Mail [email protected]

Server Requirements

Please note the following server requirements for installing the survey on your server:

The survey has been built using a series of Active Server Pages (ASP) scripts and HTML pages that use Microsoft FrontPageExtensions. The ASP scripts are written in the VBScript programming language. The survey must be run from a web server thatsupports both ASP and FrontPage Extensions. The two most common servers that support these are the Microsoft IISServer and the Microsoft Personal Web Server.

For the survey to run, Microsoft FrontPage Extensions must be loaded for the directory in which the survey is installed. Also, becausethe configuration files and the save files are in XML format, the Microsoft XMLDOM API must be available to parse and write the files.Two sets of videos have been provided. For users who complete the survey from an Apple Macintosh computer, a set of QuickTimevideos has been encoded. For users who complete the survey from a Windows PC, a set of Windows Media Videos (WMV) have beencreated. The Windows videos require Microsoft Windows Media Player version 6 or later (it is currently at version 8).

IIBrief History of Development

Beliefs-Survey Development

This instrument and the accompanying scoring rubrics were developed over a 2-year period by the authors. Six

other staff members contributed to the instrument’s development. We used a recursive cycle of development that

included designing segments, piloting them, analyzing student responses to the segments, revising the segments,

and piloting them again. We continued to consider whether our survey items were sufficient to assess each belief we

wanted to assess. The list of beliefs and the composition of the survey changed during our testing of several versions

of the survey and analysis of student responses.

The web-based survey was first used in Fall 2000 when it included one video segment with associated

questions; the platform enabled respondents to view the video in conjunction with the questions posed and to rewatch

the video if they wished. By Spring 2001, the video from the first version had been removed and three video clips had

been added to the survey. Early in 2001, all segments were in place; additional revisions were the refining of questions

related to each segment.

Five versions of the instrument were piloted on seven occasions (see table below for specific information about

administrations of the survey) so that during piloting, the survey was completed 205 times by 159 undergraduates. The

penultimate version of the survey was used at the beginning and end of a large-scale research study with 168

students, most undergraduates in their first mathematics course for teachers. For the final version of the survey, we

excluded one segment used in the large-scale study because the coding of this segment was found to be unreliable.

During piloting of the survey, we began to develop rubrics to analyze responses. Initially we categorized all

responses for each segment; we created a short characterization that captured the common features of the responses

for each category. Next we considered these characterizations for each segment in light of the belief it was designed to

assess; we ordered the categories from those that provided the most evidence for the belief to those that provided the

least evidence for the belief. While we piloted the segments and began to develop rubrics, we identified ways to adapt

the items to better elicit the beliefs to be assessed, also eliminating and adding items in the process of clarifying the

rubrics.

Most segments were in place for a pre/post pilot during the Spring 2001 semester; 29 students enrolled in the

mathematics course completed the survey at the beginning and end of the semester. Of these, 22 participated in the

treatment and the remaining 7 served as a control group. We used their responses to determine whether the survey

measured change. We compared changes in the treatment group with changes in the control group to ensure that the

survey measured the changes we expected to see as a result of the treatment. We also looked for variability

within the treatment group to ensure that the instrument captured the individual variability one would expect when 22

prospective teachers engage in the same experience.

The bulk of the instrument-development work at this point was focused on fine-tuning the rubrics, specifically

assigning scores to categories and clearly defining categories so that most responses would fit one and only one

category. One of the most important aspects of this work was testing for reliability. After a team refined a rubric and

developed a scoring system from a set of 20 responses, the team members individually applied that rubric to an

additional 30 responses to determine whether we would score them similarly. During our coding of these responses,

we encountered some coded differently by team members, so we discussed the issues raised and further clarified the

rubric.

For the purpose of validating the instrument, we asked 18 mathematics educators (7 professors and 11

mathematics education graduate students) to complete it. Their answers were coded by the coding team, and their

scores converged on the highest scores. We were convinced that our interpretations of the segments in the instrument

were compatible with those of other mathematics educators at other institutions around the country. The professors

believed the test to be a valid measure. One stated, “You can‘t observe 150 people working with children or studying

mathematics. Given the impossibility of seeing people in action to measure their beliefs, this instrument is a good

proxy” (personal communication, P. Kloosterman, June 2001).

Each belief was measured using two or three segments, so the last component of our work was to develop a

system for aggregating scores from individual rubrics to generate each overall belief score. The data used as a

reference for this task were from our pilot work. We developed our system, one that applies to each belief, mindful that

the scores are ordinal in nature. To our knowledge, this system is unique.

The survey represents countless hours of work on the part of a large team of people. We are convinced that is

the best survey possible given the constraints that we faced. The true test of its quality will be if other researchers find it

useful for their purposes. We believe that they will.

Table 1Beliefs-Survey Development

Date Participants Format Items included

Spring 200053

LiberalStudiesstudents

New-student

orientation

paper andpencil

Two autobiographical itemsSegment 3Segment 5Solving problems differently from wayteacher solves

Summer 20003 paper and

pencilSurvey, interview, experimental items

September 200015 web-based,

computer2 autobiographical itemsSegment 1Segment 2Segment 3Segment 4Segment 5One video segment (Richard)

December 200018

(15 MEFE3 210)

web-based,computer

2 autobiographical itemsSegment 1Segment 2Segment 3Segment 4Segment 5Segment 7Segment 8Segment 9

January 200133 Math 210

students(22 MEFE11 control)

17 Math 211(as post)

web-based,computer

Same as for December 2000

May 200128 Math 210

students(21 MEFE7 control)

web-based,computer

Same as for December 2000

Summer 200138 Maxonstudents

web-based,computer

Segment 7 questions

Fall 2001208 studystudents

web-based,computer

Final survey

December 2001165 studystudents

web-based,computer

Final survey

IIICoding Data

Coding Introduction

This section of the manual contains, first, the statement of the seven IMAP Beliefs the survey was designed to assess and, second, theCoding Navigator, a table with links to the scoring rubric for each belief and to the description of each segment rubric. The CodingNavigator is set up to show which rubrics are associated with each belief and to facilitate easy access to each belief rubric and to itsassociated segment rubrics. Within the table, just click on a Belief cell in the Coding Navigator to go to the manual page for the beliefscoring rubric or to a Segment cell to go to the page for the description of that segment.

The Rubrics of Rubrics follows the Coding Navigator. In the Rubrics of Rubrics document, Rebecca Ambrose explains how the scoresfor each segment were developed and then how those segment scores were combined to create a score for each belief.

Within the Belief Rubrics, a table for each belief shows how the segment scores are used to determine a score for the belief (click herefor the Belief 1 Rubric). Also, for each belief, we provide a table of the numbers and percentages of our participants who attained eachbelief score at the pre and post assessments (click here for the Belief 1 IMAP Results). These IMAP results are given for the 159 IMAPparticipants in our study.

Finally within the Segment Rubrics, we include the information coders will use for coding responses to each survey segment. Foreach segment rubric we provide (a) the Belief Statement and Rubric Description, (b) the Survey Item used in the segment, (c) anexplanation of how to determine Rubric Scores, according to the rubric, (d) a brief Scoring Summary for coders’ quick reference duringcoding, (e) Examples of responses that we have scored and the criteria on which we based each score, (f) Training Exercises andSolutions for the coders to use in learning to code the item responses with respect to the given belief, and, finally, (g) the IMAP Resultsfor that segment rubric—the numbers of our 159 participants who were assigned each of the possible rubric scores on both the pre andpost assessments. The links on this page are included so that the reader can quickly preview each element of the Segment Rubric for oneexample: B1-S3.2. Note that within the examples and training exercises for each segment rubric, the manual’s bookmarks areprovided to take one to each example and to each exercise. Therefore, coders who have access to the manual can complete the trainingexercises without having hard copies of these materials. And coders can use the bookmarks to quickly access each example or exercisefor discussion.

The Training Exercises for each segment rubric are arranged within two or three sets. The Set 1 exercises are ones that we consideredeasiest to code. The exercises in the last set are considered to be the most difficult of the examples and are lettered in reverse order. Forexample, for B1-S3.2, the Set 1 exercises are lettered A–H; the Set 2 exercises are lettered I–N; the final set, Set 3, exercises are letteredZ–U. The solutions are lettered in the same order as the exercises.

Belief About Mathematics

1. Mathematics is a web of interrelated concepts and procedures (and school mathematics should be too).

Beliefs About Learning or Knowing Mathematics, or Both

2. One’s knowledge of how to apply mathematical procedures does not necessarily go with understanding of theunderlying concepts.

3. Understanding mathematical concepts is more powerful and more generative than remembering mathematicalprocedures.

4. If students learn mathematical concepts before they learn procedures, they are more likely to understand theprocedures when they learn them. If they learn the procedures first, they are less likely ever to learn theconcepts.

Beliefs About Children's (Students') Learning and Doing Mathematics

5. Children can solve problems in novel ways before being taught how to solve such problems. Children in primarygrades generally understand more mathematics and have more flexible solution strategies than adults expect.

6. The ways children think about mathematics are generally different from the ways adults would expect them tothink about mathematics. For example, real-world contexts support children’s initial thinking whereassymbols do not.

7. During interactions related to the learning of mathematics, the teacher should allow the children to do as muchof the thinking as possible.

Beliefs B1 B2 B3 B4 B5 B6 B7

Segments S3.2 S3 S4 S3.3 S2 S2 S5

S3.3 S4 S9 S9 S5 S8 S7

S8 S7 S9

Jason Smith

Coding Navigator

IMAP Web-Based Beliefs Survey1

Rubrics-of-Rubrics Overview

Rebecca Ambrose

For our Integrating Mathematics and Pedagogy (IMAP) Web-Based Beliefs Survey, we have developed 17 rubrics associated with

seven beliefs, two or three rubrics per belief. Each rubric was based on a scale appropriate for the responses for the relevant segments of the

instrument. We began our coding by examining the range of responses in the pilot data and categorized the responses according to type. For

a given belief, we next analyzed the level of evidence for holding that belief shown in responses within each category. A response was

scored 0 if we interpreted it as showing no evidence of the belief. The highest score possible for a given item indicated strong evidence of the

belief. Rubric-scale ranges were of 3 to 5 points, depending on the range of responses and the levels of evidence we identified in the

responses The more complicated prompts resulted in a wider range of scores. Five of our rubrics have scores 0–2, 11 rubrics have scores

0–3, and 1 rubric has scores 0–4.

The scores from this instrument are ordinal in nature. A response scored 3 on the basis of a rubric for a given belief indicates that we

interpreted the response as providing more evidence for that belief than one scored 2. The difference in level of evidence between scores of 2

and 3, however, varies from rubric to rubric, from large in some cases to small in others. Because of this ordinal nature of the scores, we

cannot appropriately sum the scores and discuss means for each belief. Instead we developed a more qualitative approach to collapsing the

data to assign overall scores for each belief.

To collapse the data, we developed a rubric-of-rubrics (ROR) system that we applied to each belief. To create the rubrics-of-rubrics,

we classified each score according to the degree to which we thought that it signified evidence for the belief. In assigning these designations, 1Preparation of this paper was supported by a grant from the National Science Foundation (NSF) (REC-9979902). The views expressedare those of the authors and do not necessarily reflect the views of NSF.

we considered the particular segments and the particular scores associated with the rubrics for scoring responses to those segments. In each

case the lowest score (0) was equated with No Evidence for the belief and the highest score (which varied by rubric) was equated with

Strong Evidence. The intermediate scores were equated with either Weak Evidence or Evidence. For example, the rubric for B7-S7 has a 4-

point scale, 0–3. On this scale 0 designates No Evidence; 1 designates Weak Evidence; 2 designates Evidence; and 3 designates Strong

Evidence. Many of the 4-point scales were treated in this fashion.

Some rubrics include a Weak designation whereas others skip from the No Evidence designation to the Evidence designation. For

example, on B3-S4 scores of both 0 and 1 designate No Evidence, because responses scored 1 provided only minimal evidence for the

belief. For this item, a score of 2, however, provided more than weak evidence and so designates Evidence. The reader may wonder why

we differentiated between responses scored 0 and 1 if neither includes evidence for the belief. We designed our rubrics to capture the full

range of responses we obtained and to differentiate among them. Some responses scored 0 were extreme and provided evidence that the

respondent held a belief in opposition to the belief under consideration, whereas responses scored 1 were not extreme but still lacked

evidence for the belief. We use the rubrics not only for the rubric-of-rubrics analysis described here but also for other purposes, for which

the difference between scores of 0 and 1 score are important.

We assigned a Weak designation to responses that showed either minimal evidence for the belief or evidence in contradiction to the

belief. Some items on the instrument prompted respondents to create responses that revealed their ambivalence about the issue. For example,

in Segment 7 the respondents were asked to react to a teaching episode and then to identify the weaknesses in the teaching. Some

respondents pointed out that the guidance provided by the teacher was skillful and then, in responses to subsequent prompts, went on to

point out that the guidance provided by the teacher was excessive. We interpreted this ambivalence as weak evidence of the belief that

teachers should allow their students to do as much mathematical reasoning as possible.

After designating levels of evidence from No Evidence to Strong Evidence for each score described by each rubric, we used these

designations to develop two scoring systems to apply to our belief instrument: one system for beliefs measured using two rubrics and

another system for the beliefs measured using three segments. Each system was modified to fit the specifics of the related rubrics, for

example, to accommodate differences between 4-point rubrics that included a Weak designation and those that did not.

The scores on the rubrics-of-rubrics that are based on three rubrics range from 0–4 and are again ordinal, not interval, scores.

Respondents scoring 0 are thought not to hold the belief because they provided little or no evidence that they do. Those scoring 4 are thought

to hold the belief strongly because they provided strong evidence of the belief in the majority of responses assessing it. People receiving

scores of 1–3 showed some evidence of having the belief, but the evidence was inconsistent. For example, most people who scored 1

provided no evidence for the belief on at least one segment and some evidence in another, but the balance of the responses was toward No

Evidence. For the score of 2, the balance tipped toward Evidence, but some scores on some rubrics were for Weak Evidence. Combinations

that included at least one Strong Evidence and had Strong Evidence/Evidence scores that far outweighed the Weak Evidence/No Evidence

scores were scored 3.

One challenge we faced in developing our system was to assign scores to combinations of responses that were unexpected: responses

that showed no evidence of a belief in one segment, evidence for that belief in another, and strong evidence in a third. In the case of Belief 7,

we decided that a 0 based on one rubric indicated that the respondent did not hold the belief, so regardless of the score on the other segment,

the respondent was given a score of 0 for the belief (as indicated by the *s in Table 1). In most cases, these seemingly inconsistent scores

were for responses to items situated in mathematically different contexts. For example, the belief that one's knowledge of how to apply

mathematical procedures does not necessarily go together with one's understanding of the underlying mathematical concepts might be evident

in a response about the standard algorithm for subtraction but not in the response on division of fractions. In these cases we assigned an

overall score by assessing the strength of the belief according to the number of times it was evident. Using our system, we could treat such

cases consistently across beliefs. For example, a respondent receiving designations of Weak Evidence, Weak Evidence, Evidence for any

given belief would always receive a score of 2 for that belief.

A few anomalies in our system require justification. In some instances a 1-point change from applying a single rubric can result in a

2-point change in the rubric of rubrics. For example, scores of 0, 1, 2 for Belief 2 receive a score of 1 on the rubric of rubrics, whereas

scores of 0, 2, 2 for Belief 2 receive a score of 3, although the two sets of scores differ only by 1 point in the middle score. The middle score

derives from a segment with three scores, 0, 1, and 2, with designations of No Evidence, Weak Evidence, and Strong Evidence (no score is

associated with Evidence for this segment). The 0, 1, 2 scores thus have an NWS designation whereas the 0, 2, 2 scores have an NSS

designation. The lack of an Evidence designation for the segment gives rise to the 2-point change. Our system has few such anomalies,2 each

of which can be justified. A designation skip indicates a large difference between two scores, enough difference to warrant the skip.

0 = More Ns than anything else (except for NNS; S raises the score)

1 = ≥1 E with ≤ 1 W

2 = 1 S or 2 Es

3 = 2 E +

4 = 2 Ses

General Rubric of Rubrics

In reviewing our system, we were tempted to change a few scores, but after we revisited the data and saw the implications for changing these

scores, we were convinced that they should not be changed.

2 Our system has the following skips: Belief 2 from NWS to NSS, Belief 5 from NWW to WWE and from NWS to WES, and Belief 6from NWS to WES.

Segment-Score Designations for Each Belief

Belief 1

S3.2 S3.3

0 = N 0 = N

1 = W 1 = E

2 = E 2 = S

3 = S

Belief 2

S3 S4 S8

0 = N 0 = N 0 = N

1 = W 1 = W 1 = W

2 = E 2 = S 2 = S

3 = S

Belief 3

S4 S9

0 = N 0 = N

1 = N 1 = E

2 = E 2 = S

3 = S 3 = S

Belief 4

S3 S9

0 = N 0 = N

1 = W 1 = W

2 = E 2 = E

3 = S 3 = S

4 = S

Belief 5

S2 S5 S7

0 = N 0 = N 0 = N

1 = W 1 = W 1 = N

2 = E 2 = E 2 = E

3 = S 3 = S 3 = S

Belief 6

S2 S8, S9

0 = N 0 = N

1 = W 1 = E

2 = E 2 = S

3 = S

Belief 7

S5 S7

0 = * 0 = N

1 = N 1 = W

2 = E 2 = E

3 = S 3 = S

Note. * is explained in Table 1.

Beliefs-Scores Tables for Beliefs Scored on the Basis of 2-Rubric and 3-Rubric Combinations

Table 1Belief Scores for 2-Rubric CombinationScores

Evidence levels Belief score

*N, *W, *E, *S 0

NN 0

NW 0

NE 1

NS 2

WW 1

WE 1

WS 2

EE 2

ES 3

SS 3Note. * is a score of 0 on S5 for Belief 7.

Table 2Belief Scores for 3-Rubric CombinationScores


NNN 0

NNW 0

NNE 1

NNS 1

NWW 1

NWE 1

NWS 2

NEE 2

NES 2

NSS 3

WWW 1

WWE 2

WWS 2


WEE 2

WES 3

WSS 3

EEE 3

EES 3

ESS 4

SSS 4

III.5Belief Rubrics

Belief 1 Rubric

S3.2 S3.3Segment Score Segment Score

0 01 12 23

BeliefScore

Permutations of(S3.2, S3.3)

0 (0,0) (1,0)

1 (0,1) (1,1) (2,0)

2 (0,2) (1,2) (2,1) (3,0)

3 (2,2) (3,1) (3,2)

Scoren % n %

0 95 60% 30 19%1 47 30% 64 40%2 15 9% 39 25%3 2 1% 26 16%Total 159 159

Pre Post

IMAP Results for Belief 1

Belief 2 Rubric

S3 S4 S8Segment Score Segment Score Segment Score

0 0 01 1 12 2 23

BeliefScore

Permutations for(S3, S4, S8)

0 (0,0,0) (0,0,1) (0,1,0) (1,0,0)

1 (0,0,2) (0,1,1) (0,2,0) (1,0,1) (1,1,0) (1,1,1) (2,0,0) (2,0,1) (2,1,0) (3,0,0)

2 (0,1,2) (0,2,1) (1,0,2) (1,1,2) (1,2,0) (2,0,2) (1,2,1) (2,1,1) (2,2,0) (3,0,1)(3,1,0) (3,1,1)

3 (0,2,2) (1,2,2) (2,1,2) (2,2,1) (3,0,2) (3,1,2) (3,2,0) (3,2,1)

4 (2,2,2) (3,2,2)

Scoren % n %

0 122 77% 63 40%1 23 14% 31 19%2 11 7% 34 21%3 3 2% 23 14%4 0 0% 8 5%Total 159 159

Pre Post


Belief 3 Rubric

S4 S9Segment Score Segment Score

0 01 12 23 3

BeliefScore

Permutations for(S4, S9)

0 (0,0) (1,0)

1 (0,1) (1,1) (2,0)

2 (0,2) (0,3) (1,2) (1,3) (2,1) (3,0)

3 (2,2) (2,3) (3,1) (3,2) (3,3)

Scoren % n %

0 102 64% 44 28%1 20 13% 18 11%2 29 18% 41 26%3 8 5% 56 35%Total 159 159

Pre Post


Belief 4 Rubric


0 01 12 23 34

BeliefScore


0 (0,0) (0,1) (1,0)

1 (0,2) (1,1) (1,2) (2,0) (2,1)

2 (0,3) (1,3) (2,2) (3,0) (3,1) (4,0) (4,1)

3 (2,3) (3,2) (3,3) (4,2) (4,3)

Scoren % n %

0 114 72% 57 36%1 36 23% 56 35%2 9 6% 22 14%3 0 0% 24 15%Total 159 159

Pre Post


Belief 5 Rubric


0 0 01 1 12 2 23 3 3

BeliefScore


0 (0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1)

1(0,0,2) (0,0,3) (0,1,2) (0,2,0) (0,2,1) (0,3,0) (0,3,1) (1,0,2) (1,1,0) (1,1,1)(2,0,0) (2,0,1) (3,0,0) (3,0,1) (1,2,0) (1,2,1) (2,1,0) (2,1,1)

2(0,1,3) (0,2,2) (0,2,3) (0,3,2) (1,0,3) (1,1,2) (1,1,3) (2,0,2) (2,0,3) (3,0,2)(3,1,0) (3,1,1) (3,2,0) (3,2,1) (1,2,2) (1,2,3) (1,3,0) (1,3,1) (2,1,2) (2,1,3)(2,2,0) (2,2,1) (2,3,0) (2,3,1)

3(0,3,3) (3,0,3) (3,1,2) (3,2,2) (3,1,3) (3,3,0) (3,3,1) (1,3,2) (1,3,3) (2,2,2)(2,2,3) (2,3,2) (2,3,3)

4 (3,2,3) (3,3,2) (3,3,3)

Scoren % n %

0 53 33% 25 16%1 72 45% 55 35%2 28 18% 45 28%3 5 3% 28 18%4 1 1% 6 4%Total 159 159

Pre Post


Belief 6 Rubric


0 0 01 1 12 2 23

BeliefScore


0 (0,0,0) (1,0,0)

1(0,0,1) (0,0,2) (1,0,1) (2,0,0) (3,0,0) (1,1,0) (0,1,0)(0,2,0)

2(1,0,2) (2,0,1) (2,0,2) (2,1,0) (2,2,0) (3,0,1) (1,1,1)(1,2,0) (0,1,1) (0,1,2) (0,2,1) (3,1,0)

3(2,1,1) (2,1,2) (2,2,1) (3,0,2) (1,1,2) (1,2,1) (1,2,2)(0,2,2) (3,1,1) (3,2,0)

4 (2,2,2) (3,1,2) (3,2,1) (3,2,2)

Scoren % n %

0 65 41% 28 18%1 57 36% 56 35%2 29 18% 40 25%3 7 4% 27 17%4 1 1% 8 5%Total 159 159

Pre Post


Belief 7 Rubric


0 01 12 23 3

BeliefScore


0 (0,0) (0,1) (0,2) (0,3) (1,0) (1,1)

1 (1,2) (2,0) (2,1)

2 (1,3) (2,2) (3,0) (3,1)

3 (2,3) (3,2) (3,3)

Scoren % n %

0 113 71% 64 40%1 39 25% 58 36%2 7 4% 30 19%3 0 0% 7 4%Total 159 159

Pre Post


III.6Segment Rubrics

B1-S3.2Rubric for Belief 1—Segment 3.2

Belief 1

Mathematics is a web of interrelated concepts and procedures (and school mathematics should be too).

Description of Rubric

We operationalized this belief by thinking of a web of interrelated concepts and procedures as consisting of nodesand the connections among them. For this rubric, the nodes are the five strategies whereas statements about therelationships between strategies represent the connections between nodes. The number of strategies arespondent chooses to share is thus one factor in determining a respondent’s score. For example, if a respondentchooses to share two or fewer strategies, she receives the lowest score on the rubric, regardless of herexplanations for wanting to share the strategies. We reasoned that even if a respondent makes a connectionbetween two strategies, she still does not value the web-like nature of mathematics if she has chosen to share onlytwo strategies. Therefore, to receive higher scores on the rubric, respondents must choose to share three or morestrategies. For example, to receive the second-highest score on the rubric, a respondent must choose to shareeither four or five strategies and must value all or most of them. Further, to receive the highest score on the rubric,a respondent must choose to share ALL the strategies, make comments to indicate that she values each strategy,and make at least one explicit connection among strategies. This last criterion (that a respondent makes an explicitconnection among strategies) addresses the interrelationship between concepts and procedures (or theconnections between the nodes).

Carlos149 + 286

Written on paper

Henry149 + 286

Henry says, "I know that 40 and 80 is 120, and one hundred and two hundred makes 300, and 120 and 300 is 420, and 9 and 6 is 14, so 420 and 10 is 430, and 4 more is 434."

Elliott149 + 286

Written on paper

Sarah149 + 286

Sarah says, "Well, 149 is only 1 away from 150, so 150 and 200 is 350, and 80 more is 430, and 6 more is 436. Then I have to subtract the 1, so it is 435."

MariaManipulatives

= 100Called a flat

= 10Called a long

= 1Called a single

Maria uses manipulatives (base-ten blocks) to solve the problem. Maria says, "I took one flat for the 100 in 149 and 2 flats for the 200 in 286.

I took 12 longs: 4 for the 40 in 149 and 8 for the 80 in 286.

I took 15 singles for the 9 in 149 and the 6 in 286.

Then I counted like this, '100, 200, 300'; then for the longs, '310, 320,

3.2. If you were a teacher, which of the approaches would you like to see children share? Select "yes" or "no" next to each student and then explain why or why not.

Carlos Yes

No

Henry Yes

No

Elliott Yes

No

Sarah Yes

No

Maria Yes

No

3.3 Consider just the strategies on which you would focus in a unit on multidigit addition. Over a several-weeks unit, in which order would you focus on these strategies?

330, 340, 350, 360, 370, 380, 390, 400, 410, 420'; then the singles, '421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435.'

So the answer is 435."

First:

Second:

Third:

Fourth:

Fifth:

Please explain your answer for the rankings in 3.3 above.

3.4. Do you think that Carlos could make sense of and explain Sarah's strategy? Why or why not?

3.5. Do you think that Carlos could make sense of and explain Elliott's strategy? Why or why not?

Check your answers and then click on the Submit button:

B1-S3.2Rubric Scores

0. Responses scored 0 indicate no appreciation for the interconnectedness of procedures and concepts. A decision to shareonly one or two strategies is interpreted as evidence of this lack of appreciation. Such respondents tend to express a definitepreference for Carlos's strategy.

1. Responses scored 1 indicate little appreciation for the interconnectedness of concepts and procedures. Despite the numberof strategies the respondent wants shared, the supporting explanations give little or no indication that the strategies areconnected. The explanations for wanting to share tend to show little appreciation for having multiple approaches to solveproblems. Many people who score 1 also have a definite preference for Carlos’s strategy.

2. Responses scored 2 indicate a valuing of multiple ways to understand big ideas in mathematics. They may indicate thatsome procedures and concepts are interconnected, although often the connections are not explicit. Most or all strategies inaddition to Carlos's are valued, and reasonable appreciation tends to be expressed for the strategies. However, a score of 2differs from 3 because those scoring 2 tend not to write about the interconnectedness of the strategies.

3. Responses scored 3 indicate appreciation for the interconnectedness of concepts and procedures and valuing of multipleways to understand big ideas in mathematics. Respondents would have students share all strategies, and they tend to havemathematically appropriate reasons for the sharing; they indicate that they value all strategies. Some of their explanationssupport the idea that mathematics is a web. That is, students who score 3 write about the connections between (and among)at least some strategies. A response scored 3 differs from one scored 2 because of the explicit mention of the mathematicalconnections among strategies. Also, one scoring 3 does not state a preference for Carlos's strategy.

B1-S3.2

Scoring Summary

Score Rubric details

0 • Shares 1 or 2

1• Shares 3 OR• Shares 4, AND 1 or more explanations for sharing strategies are negative overall.• Shares 5, AND 2 or more explanations for sharing strategies are negative overall.

2

• Shares 4 AND values all the strategies selected to be shared.• Shares 5 AND EITHER

o does not discuss any connections among strategies ORo has a response about one strategy that is negative overall.

3 • Shares 5, values all strategies, and makes at least one explicit and appropriate mathematical connectionbetween or among the strategies.

Comments on Scoring

*Note one exception to the number of students who share: A participant states that Carlos's strategy should not beshared. For purposes of using the rubric, we count this preference not to share Carlos’s strategy as a shared strategy(Carlos exception), because most reasons for choosing not to share Carlos’s strategy show evidence of this belief.Similarly, do not count as negative any responses that indicate that Carlos may not understand his strategy.

B1-S3.2Examples

1qCarlos_share

q3c_Carlos_whyshare

qHenry_share

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qElliott_share

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qSarah_share

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q3c_Maria_whyshare Score Comment

Yes It is the rightway andeasiest way ifthe childrenunderstand it.

No This way istoocomplicatedwith too manynumbers.The childrenwould getconfused andmess up likeHenry did.

No No, becausehe did itbackwardsfrom left toright and it'sharder tounderstandthe processthat way.

No Sarah knewwhat she wasdoing but thisway has moresteps thanneeded.

Yes This way showsthe child exactlywhat they areadding.

0 Respondent haschosen only tworesponses to beshared, thus theexplanations haveno effect on thescore.

2qCarlos_share

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Yes Visual learnerscould learnfrom this.

No May be a littletoo confusingfor others.

No There is norealexplanationand visuallearners maybe confused.

Yes This is a verygoodexplanationwithout beingconfusing.

Yes Hands-onlearners wouldbe able to graspthis withoutmajor problems,beacuase theycan see theproblem laidactually laid outfor them.

1 Respondent haschosen threeresponses to beshared, thus theexplanations haveno effect on thescore.

3qCarlos_share

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Yes It shows thathe knows howto add largernumbers verywell

No Understanding like thisconfuses meand if i am theteacher i can'timagine whatit is doing tothe otherchildren

Yes For me this issomethingthat i can seehis thinkingand help himcorrect mucheasier than ifhe rattled offnumbers andi had to tryand follow histhinking

Yes I would behappy thatshe didadding welland that sherememberedto subtractthe one thatshe added atthe beginning

Yes She has astrong base ofplace value andi can see histrain of thoughtand what he isdoing to get theanswer

1 Shares four, withone overall negativeresponse. Thereason for sharingElliott’s strategyindicates that therespondent does notappreciate Elliott'sreasoning butinstead would try tocorrect it.

B1-S3.24qCarlos_share

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qHenry_share

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qSarah_share

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Yes This is the wayI was taught.

Yes. This makessense buttakes toolong. It wouldbe easy tomakemistakes

Yes This alsomakes sense,but I thinkthere is toomuch roomfor error.

Yes This is logicaland showsgood numbersense.

Yes This is logical 1 Shares five, with twooverall negativeresponses (forHenry’s and Elliott’sstrategies).

5qCarlos_share

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Yes This is themost commonand easierway.

No I feel they'llget confusedand couldmake amistakeeasier.

Yes This will showme that theyreallyunderstandthe concept ofmath. Byadding thehunderdstogether, thetens and theones, andthen a finaladdition.

Yes I wouldn'tmind mystudentsrounding uptheir answersand thentaking awaythatdifferencethat they hadused to makethe problemeasier. Itcould be alittleconfusing andchallengingtrying to addand subtractbut if they cando it, then Iwouldn'tmind.

Yes Some childrenneed visuals forbetterunderstandinginstead ofnumbers. Ifusingblocks/manipulatives could helpthem have abetterunderstandingthen it's finewith me.

2 Respondent sharesfour and allexplanations showreasonableappreciation forhaving multiple waysto solve problems.


q3c_Carlos_whyshare

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'Yes 'That is thetypical way Ihave seen tosolve addingproblems. It isalso efficientand easy forthem tounderstand

'Yes 'shows goodnumbersense

'Yes 'shows goodplace valueunderstanding

'Yes 'shows goodnumbersense

'Yes 'shows goodplace valueunderstanding

2 Shares five, valuesall, but makes noconnections.

7qCarlos_share

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'Yes This is themost simpleway to solve it.It is just easymath addedstraight up anddown.

'Yes His method isvery good. Itshows adifferentdirectgin toansering theproblem. Analternativeapproach tosolving thisproblem mayappeal toother childrenwho may bestruggling toanswer.

'Yes This methodmay serve toconfuse morechildren thanit helps.Elliot’smethod takesa little morethinking butthe risk ofmore errorsincreaseswhen ther aremore steps.

'Yes 'Sarah easilyuses roundernumbers.Her method iseasily shownandexplained.

'Yes 'This methodhelps children toactually seewhat they areadding.

2 Shares five, has oneshared responsethat is negativeoverall (for Elliott’sstrategy).


q3c_Carlos_whyshare

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Yes This is thetraditionalalgorithm,which childrenare likel tohave seen inschool or athome. Itprovides anelegant andconcise way torecord aprocedure.Doing theprocedurecorrectly doesnot ensureunderstanding,but havingCarlos presentthe procedurewould providean avenue forconnecting hismethod,Elliott'smethod, andMaria'smethod.

Yes Henry ismaking use ofplace valueand facts thathe knows inorder to solvethe problem.His solutionshows thatyou do nothave to starton either theleft or theright.Breaking 14into 10 and 4shows someplace valueunderstanding.

Yes Elliott'smethodshows anunderstanding of placevalue and thecommutativityof addition. Itis a methodthat is likely tobe easilyaccessible tootherchildren. Itconnectsnicely toCarlos, Henryand Maria'smethods.

Yes Sarah'smethod maynot beunderstoodby all of theotherchildren, but Iwould stillwant her toexplain itbecause itwould makesense tosomechildren. Itshows amental mathway ofthinking aboutthe problemby usingcompensation. I wouldwant otherchildren tosee theflexible way ofthinking thatis involved.

Yes Maria's solutionprovides aconcreterepresentationof the placevalues and canbe inspectedfrom severalangles. It couldbe used toaugmentCarlos, Henry,and Elliott'ssolutions bydoing thecounting indifferent ways.Maria's methodis bothkinesthetic andvisual andprovides aconcreterepresentationof the problemand thesolution.

3 A strong 3 becauseall five strategies arechosen and highlyvalued, and multipleconnections aremade amongstrategies.


q3c_Carlos_whyshare

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Yes I think this isthe mostcommon wayto do math.This is one ofthe methods Iwould beteaching theclass, so hecould help meexplain it.

Yes Henry's wayis a great fastway to domath in yourhead or evenorally, andeven thoughhe made amistake Ithink thestudentswould beinterested tosee how hedid theproblem.

Yes I think verysimilar toHenry’s wayonly writtendown so Iwouldprobably havethen presenttogether andthen theycould seehow oneanother didthe problem,and howsimilar theyare.

Yes Yes becauseI think shehas thegreatestunderstanding of math.Sheunderstoodthat you canmanipulatethe numbersto get them towork with youhowever youwanted. Andthen you cancontinue to goback andmakechanges evenafter you areclose to ananswer.

Yes Maria’s way isan excellentexample of howto usemanipulative tosolve mathproblem and Iwould want mystudents toknow that theycan always usemanipulative tohelp them ifthey areconfused.

3 Typical 3 becauseall five strategies areto be shared, atleast onemathematicalconnection is madebetween strategies(Elliott’s andHenry’s), and clearlyall the strategies arevalued.

10qCarlos_share

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'No 'by the chldlearning thisprocess theymay beworking it outand notunderstandwhat they aredoing.

'Yes 'the child isusing benchmarks tosolve theproblem.

'Yes the child isable to seethe placevalues andclearlyunderstandsthe problem

'Yes 'Like Henry,the child isusingbenchmarksand clearlyunderstandswhat they aredoing

'Yes using thismethod thechild is able tosee exacltywhat they aredoing helpingthemunderstand

3 Carlos exception.Although only fourresponses areselected to beshared, Carlos’s notbeing selectedis—for purposes ofusing therubric—counted as ashared strategy.Thus, shares five,makes a connectionbetween Henry’sand Sarah’s, andvalues all strategies.

B1-S3.2Training Exercises—Set 1

AqCarlos_share

q3c_Carlos_whyshare

qHenry_share

q3c_Henry_whyshare

qElliott_share


qSarah_share

q3c_Sarah_whyshare

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Yes it is an easiermethod forchildren tolearn by and agood way tostart offlearning.

No I don't believehis methodwouldcomprehendible to otherchildren.

No he did it in away that gavethe rightanswer, butthe methodhe usedshouldn'talways beused forproblemsolving.

No she just madethis problemmuch moredifficult for herto solve thanit was. sheused to manydifferentnumbers forhertechninque.

Yes Maria's methodwas greatbecause it isdifferent and itallows otherchildren to learnin a morecomplicated, yetunderstandingway.

BqCarlos_share

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qHenry_share

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qSarah_share

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Yes This is thetraditional waythat teachersteach theirstudents how toget the correctanswer from thistype of problem.I would like tohear how hethought aboutthis problem andif he really knowswhat the carryingof the ones isreally about.

Yes This is a greatway to solve aproblem quicklyand easily inhishead. Iwould want himto share thiswith the classso perhapsstudents couldalso lean hismethod.

Yes This is alonger versionof thetraditionalalgorithm. Butis is a goodway becauseeach step isshown andwhencompared toCarlos' way ofthinking youcan actuallysee what goeson behind thescenes.

Yes Much likeHenry's way ofthinking, thismethod can beused in theoutside world.Without havingto use a paperand pencil andblocks to showhow they cameup with theanswer, theygot the correctanswer bysimplyestimating andusing theirbrains.

Yes This method is agood visual wayof seeing how themathematicsworks. It is goodwhen justlearning theconcepts and forlearning how tocount by 10's and100's. I wouldreally encourageher to share herthinking with theclass.

B1-S3.2CqCarlos_share

q3c_Carlos_whyshare

qHenry_share

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qSarah_share

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Yes it does showthe algorithm,but notnecessarilyanyunderstanding

Yes even thoughthere is anerror, it showsthinkingmathematiacally

Yes but a lot tokeep in yourhead

Yes but a lot tokeep in mind

Yes seems tediousto count by tens

DqCarlos_share

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Yes Carlos learnedthe traditionalway andcorrectlyregroupedwhen needed

Yes Henry'sstrategy isgood but hejust messedup on hisaddtion withadding the 6and 9. heshowed deepthought andregrouping

No Elliott workedbackwardsand startedfrom thebottom, youlook at theproblem andat first say itswrong butthen youunderstandwhat she isdoing. i thinkit can confusechildren theway he set itup.

Yes Sarah isusingrounding andher numbersense togroup likeamountstogetherwhich isgood. i likethat she seespairs ofnumbers thatwould addtogethernicely

yes Maria has anunderstandingof numbers andplace value.she showedwhat shestarted off withand regroupedfor what sheneeded to do. ithink she reallyunderstood theproblem betterbecause shevisually workedthe problem out

EqCarlos_share

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Yes shows workand methodclearly

Yes method isclear

No it works but itmight confuseother studentsince it is notthe traditionalway

Yes method clear Yes method clear

B1-S3.2FqCarlos_share

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qSarah_share

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Yes It's the basicprocedure foraddition.

No Solving aproblem insentenceformat can bedifficult tofollow.

Yes His way ofthinking isright andunderstandable to me andprobably tomost childrenwho arescared ofadding bignumbers.

No Thisprocedure,althoughunderstandable, can lead toerror becauseyou have toadd andsubstractcertainamountswhich may, inthe end, beforgottenabout.

Yes This is a visiualway to add andit may helpothers to learnthe procedure athand

GqCarlos_share

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qSarah_share

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No this approachseems to justbe the waysomeonetaught him todo it. theredoe not seemto be muchunderstandingof what isactuallyhappening.

No for the mostpart Henry'sanswer isright andshows heunderstandswhat he isdoing.however hemisscalculatesand thereforehas the wronganswer.

Yes it is similar toHenry's buthe does moreof the work inhis head, butit is a goodmethod ofsolving theproblem

Yes i think sheshows themostunderstanding and it is agood way tolook at theproblem andit would alsobe easy forthe rest of theclass tounderstand

Yes this is a verygood visual ofthe problem andsome childrenmay understandthe problemnore clearly ifhe showed it inclass

B1-S3.2HqCarlos_share

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qSarah_share

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Yes This is a verygoodapproach.

Yes This is a goodapproach andI would havethe classcorrect theerror.

Yes I think this isa niceapproach tounderstandplace value.

Yes I think that if itwill helpanother childthen I thinkshe shouldshare herapproach

Yes This approachhelps a childvisualize theproblem


IqCarlos_share

q3c_Carlos_whyshare

qHenry_share

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qSarah_share

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'Yes 'This is a quickand easy wayof addingmultiple- digitnumbers, if themethod isdone correctly.

'Yes 'Henry hasshown hisunderstanding of ones,tens, andhundreds bysimplifying itand breakingit down intoseveralsmallerproblems.Although hemiscalculatedthe last digit,it is still agood conceptto use.

'No 'I do not thinkthis is a verygood methodto be using. Ibelieveinstead thatthe differentmath Elliottdid, shouldhave beenseparated. Ibelieve all oftheseseparateadditionscompiled intoone, wouldend up beingconfusing andmore difficult.

'Yes 'I like howSarahsimplified theaddition tomake it easierfor her tofigure out. Aslong as shedoesn't"round off"too manynumbers, Ibelieve this isa good way tofigure out thisproblem.

'Yes 'I believemodels arealways a goodtool for mathproblems. Thisway a child canvisualy see themath takingplace. Countingblocks may takelonger, but willallow children tohave a betterunderstandingof multi-digitaddition.

JqCarlos_share

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'Yes I would like tosee Carlosshare hismethodbecauseCarlos showsunderstandingof place value,and he does itin the shortway. I feel thatit would bebeneficial forother childrento see thismodeled tothem.

'No 'A child couldvery easilyget confusedwith thismethod.Henry himselfgot confusedas hemiscountedwhen hemade the 9and the 6 into10 and 4.

'Yes 'This methodclearly showshow Elliot wasthinking and Ithink that fromElliots methodthe childrenwould have abetterunderstandingof place value,from the partwhere Elliotadded the 40and the 80.MAny childrenwould add thenas units and nottens.

'No 'Again this isa veryadvancedmethod, but ifthere are anyweak childrenin the class ifeel that theywouldntbenefit fromthis method, Ifear that itwould justconfuse them.

'Yes 'If many of thechildren didntunderstandplace value thenthis is anexcellent way toexplain thisproblem tothem. This waythey are linkingthe writtensymbol to thepicture (model)which isimportat.

B1-S3.2KqCarlos_share

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'No 'I like Carlos'methodbecause it isvery traditional.he is carryingthe 1's to thenext place. iwould alsowant to clarifythat thestudent knowswhat thoseonesrepresent. Forexample the 1carried into thesecond columactually is a10.

'Yes 'Although ibelieve thismethod canget confusing,it shows placevalue verywell. heknows thatthe 1 in 149represents100 and notjust one. ibelieve Henryhas goodunderstanding and i thinkthis couldtranslate tothe rest of theclass.

'Yes 'Elliot andHenry hadabout thesamemethod, butElliot wrotehis answerdown. i thinkit would benice for thestudents toactually seethe problemwritten downlike thisinstead of justhearing theexplanationfrom Henry.

'Yes 'Somestudents mayunderstandthis methodwell. it iseasier forstudents towork withcleannumbers thatend in 0 or 5.i think this is agood way tosolve theproblem. Theonly thing thatwould worryme is if Sarahforgot tosubtract the 1at the end.

'Yes 'Manupulativesand visual aidsare very forcefulwhen teachignstudents. ibelieve that bybeing able toactually see theproblem withblocks that arerelative to thesize of thenumber, theyare able tounderstand.Themanipulativeblocks also helpchild

LqCarlos_share

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'Yes 'His is verysimple andmost kids canadd two andthree digitnumbers ifthey learn howto add 1 digitnumberstogether

'No 'Henry is toowordy andsome kidsmight getconfused byall the wordsand numbers.

'Yes 'His is alsovery simpleand for thosewhounderstandCarlos theywouldprobably beable to seehow Elliot didit.

'No 'Sarahs isalso to wordyand has toomanynumbers in it

'No 'Once again toowordy. Alsokids like to keepit as simple aspossible and ifthey saw thisthey might getintimidated byall the graphsand numbersand words.

B1-S3.2MqCarlos_share

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'Yes Give thestudentsanother way oflooking atplace value.

Yes This is a littlemorechallenging,but I wouldprobablyteach this tothem after afull andcompleteunderstanding of placevalue

'Yes This methodalso gives analternativeway oflooking at andsolving placevalueproblems, thiswouldprobably runside by sidewith Carlos.

Yes This will notbe taught untillater in thelesson or notat all. I feel itis too difficultfor children tolearn at sucha young age.

'No Toocomplicated!!

NqCarlos_share

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q3c_Sarah_whyshare

qMaria_share


'Yes This is the wayI was broughtup how tosolve thisproblem andthink it isn’tdifficult at all.

'Yes His reasoningis a little morecomplex, youreally have tothink about it,but it makes alot of sense,and whoknows,maybe a childwould like thismethod more.

'Yes This isanotherinterestingway thatchildren cansolve theproblem. Iwasnevertaught thisproblem but itmakes perfectsense and Ithink thechildrenshould beopen to lots ofways on howto solveproblems.

'Yes 'Although thisis one way tosolve theproblem, Ithink it wouldbe tooconfusing forchildren tolearn.

'Yes I think this oneshould be usedbecausechildren arevisual learnersand shouldhave theopportunity towork with baseten blocks sothat they canactually see theproblem.


ZqCarlos_share

q3c_Carlos_whyshare

qHenry_share

q3c_Henry_whyshare

qElliott_share


qSarah_share

q3c_Sarah_whyshare

qMaria_share


Yes Carlos did itthe simpliestand quickestway and gotthe correctanswer.

No Henry is theonly one whogot theanswerwrong. heuses toomany stepsalso.

No Elliot gets thecorrectanswer buthis way ismuch longerthan Carlos'

Yes Sarah usesgoodreasoning. ayoung studentmight forgetto subtractthe ONE inthe endhowever. ithink thismethod wouldbe OK for aolder studentwho knowshow to roundup or downand write outall of thesteps. its toocomplicatedthough foryoungerstudents.

No although Mariagets the answerright in the endhis method isthe longest andmostcomplicated.

YqCarlos_share

q3c_Carlos_whyshare

qHenry_share

q3c_Henry_whyshare

qElliott_share


qSarah_share

q3c_Sarah_whyshare

qMaria_share


Yes it's the mostaccepted wayand is easywith large orsmall numbers

Yes shows goodunderstanding of placevalue

Yes goodunderstanding of placevalue

Yes goodmanipulationof theproblem

Yes good practice

B1-S3.2XqCarlos_share

q3c_Carlos_whyshare

qHenry_share

q3c_Henry_whyshare

qElliott_share


qSarah_share

q3c_Sarah_whyshare

qMaria_share


Yes Definitelybecause thischild knowsthe shortcutsand is fartherahead of theother studentswhen it comesto doing thework in hishead.

Yes I didn't reallyunderstandthis child'sway ofapproachingthe problemand maybe if Ihad a secondopinion Imight be ableto analyze thedescription alittle betterand explain tothe childrentwhy Henrygot theanswerwrong.

Yes I would likeElliott toshare hisdescriptionbecause hehas all of hisaddition andsubtractionfigured outnow he justneeds tolearn theshortcuts.

Yes Sarah alsohas heraddition andsubtractionfigured outpretty well.Now she justneeds tolearnshortcuts.

Yes I really think thischild haspotential inlearningsubtraction andaddition he justtook theextremely longway inapproaching theproblem. Hestill has a littleway to gobefore learninghow to add andsubtract but thismethod teacheschildren how tocount.

WqCarlos_share

q3c_Carlos_whyshare

qHenry_share

q3c_Henry_whyshare

qElliott_share


qSarah_share

q3c_Sarah_whyshare

qMaria_share


Yes It is probablythe mostacceptedanswer andwith otherclasses and sowould be goodto show.

Yes Even thoughhe got itwrong itwould begood to showhis with Elliotsto who theway theydiffered andsee if hecould catchhis ownmistake

Yes I don’t thinkmany kidsthink like thisand so itwould be agood way toshow andHenry thoughtin the sameway andwould begood tocomparethem

Yes This was agood way tome it showedherunderstanding the #s andsharig withthe classwould let thekids seedifferent wayof thinking

Yes Working withblocks is a goodway as well andit gives the kidsa visual sense,and I think thiswould be agood wayt toshare with theclass.

B1-S3.2VqCarlos_share

q3c_Carlos_whyshare

qHenry_share

q3c_Henry_whyshare

qElliott_share


qSarah_share

q3c_Sarah_whyshare

qMaria_share


Yes it is the fastestway and is themost accepted.

No it doesn'tseem to haveany logic orreasoning. itseems like itis just addingthinds thatlook like theyshould gotogether.

Yes if given thecorrectexplanation ofthis method ithink that itcould workwell to teachchildren placevalue.

Yes as long asthey knowthat this is amore difficultway to thinkabout it.

Yes it gives anexcellentexample of thevalue of eachplace and itgives a visualexample.

UqCarlos_share

q3c_Carlos_whyshare

qHenry_share

q3c_Henry_whyshare

qElliott_share


qSarah_share

q3c_Sarah_whyshare

qMaria_share


'Yes 'clear, simple,to the pointand seems themost logicaland easy tounderstand,especially forother students

'No 'first, he wasone off for theanswer andhis logic maymake senseto him but beconfusing forotherstudents

'No 'if you weregoing to havethem shareideas i thinkCarlos'sproblemwould be abetter methodfor doing itthis way.Elliottsmethod maybe ratherconfusing forotherstudents. ican see whathe did butCarlos's wasmore simpleand i think"easier" tound

'Yes 'this is a reallygood way ofdoing itlogically inyour head,especially ifyou don'thave scratchpaper.

'Yes 'this would be agood way ofsolving theproblem forstudents whohad a hard timeseeing theproblemconceptually.this method sortof lays it outvisually forthem.

B1-S3.2Solutions for Training Exercises

Exercise Score Comment

A 0 Share only two strategies. No need to look at explanation.

B 3Share all five strategies, all strategies clearly valued, and at least one connection betweenstrategies (in this case, the connection is between Henry's and Sarah's strategies) made.

C 1Although all five strategies shared, the explanations for Elliott, Sarah, and Maria indicatethat the respondent may not appreciate these strategies or appreciate having multiple waysto solve problems.

D 2 Share four strategies, all valued.

E 1Share four strategies; explanations too brief to judge whether the respondent has anappreciation for the multiple ways to solve problems.

F 1Share exactly three strategies. No need to read the explanations because the respondentshared three.

B1-S3.2Exercise Score Comment

G 2

Share three, but not Carlos's strategy. With the Carlos exception (see *), this response isscored 2 (i.e., we count this as sharing four strategies). The other explanations indicate areasonable appreciation for the strategies that are shared and overall valuing of multipleways to solve problems.

H 2Share all five strategies, reasonable appreciation for all strategies, but no connectionsamong strategies stated.

I 2 Four strategies are chosen and appreciation is shown for all four.

J 1 Shares three.

K 3

Carlos exception. Although only four strategies are selected to be shared, Carlos’s was notselected, so for purposes of using the rubric, we count this response as sharing of five;respondent values all responses overall and makes a connection (between Henry's andElliott’s strategies).

L 0 Shares two.

M 1Shares four, but one explanation (for Sarah’s) is negative overall. Note that a connection ismade between Elliott’s and Carlos’s strategies, but because only four are chosen to beshared (and one explanation is negative), this response is scored 1.

N 2 Shares all, but has one negative overall explanation (for Sarah’s strategy).

B1-S3.2Exercise Score Comment

Z 0 Share only two strategies; no need to look at explanations.

Y 2Share all five strategies, strategies to be shared are valued; appreciation for having multipleways to solve problems shown, but connections between or among strategies not statedexplicitly.

X 1Share all five strategies, but definite preference for Carlos's strategy. The other strategiesare discussed as a means to learning Carlos's strategy.

W 3Share all five strategies, all valued; a connection made between Henry's and Elliott'sstrategies.

V 1Although four strategies are to be shared, respondent does not show appreciation forSarah's strategy and qualifies sharing Elliott's strategy.

U 1 Exactly three strategies are to be shared.

Scoren % n %

0 53 33% 7 4%1 75 47% 41 26%2 29 18% 87 55%3 2 1% 24 15%Total 159 159

Pre Post

IMAP Results for Belief 1 Segment 3.2

B1-S3.3

Rubric for Belief 1—Segment 3.3

Belief 1

Mathematics is a web of interrelated concepts and procedures (and school mathematics should be too).


Respondents are explicitly asked to select an order in which they would focus on strategies if they were to teacha place-value unit, and then they are asked to explain their order. Because they are asked to provide oneexplanation for their order (instead of explaining each strategy in turn), the item lends itself to writing aboutconnections among and between strategies. So, if respondents make no connections between or amongstrategies, they receive the lowest possible score on this item. Furthermore, to receive the highest score on theitem, a respondent must choose to share at least four strategies AND make at least two statements thatexplicitly connect the strategies with one another. We take these connecting statements as proxies for therespondent’s belief about the interrelatedness of concepts and procedures.

We also look for conceptual statements made about the individual strategies to identify respondents who mayrecognize the conceptual benefits of a strategy but do not explicitly connect those strategies to other strategies.Respondents who have two such responses receive the middle score. Within this item, we score the web-likeaspect of mathematics by examining the number of strategies on which the respondent chooses to focus whileteaching the place-value unit. Although respondents could choose to share all five strategies and still receivethe lowest score because of the lack of mathematical connections mentioned in their explanations, scores arecapped based on the number of strategies selected. For example, if respondents choose to focus on one or twostrategies, they receive the lowest score; if respondents choose to focus on exactly three strategies, they receiveeither the lowest or middle score; and if the respondents choose to focus on four or five strategies, their scoresare based on their explanations.

Carlos149 + 286

Written on paper

Henry149 + 286


Elliott149 + 286

Written on paper

Sarah149 + 286


MariaManipulatives

= 100Called a flat

= 10Called a long

= 1Called a single






Carlos Yes

No

Henry Yes

No

Elliott Yes

No

Sarah Yes

No

Maria Yes

No


330, 340, 350, 360, 370, 380, 390, 400, 410, 420'; then the singles, '421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435.'


First:

Second:

Third:

Fourth:

Fifth:





B1-S3.3

Rubric Scores

0. Responses scored 0 do not indicate the belief that mathematics is an interrelated web of concepts and procedures. Inparticular, they do not mention connections among strategies, nor do they mention the concepts related to particularstrategies. Respondents may indicate that they chose an easiest-to-most-difficult progression, without stating why onestrategy is easier than another, or they may make no conceptual connections within or among the strategies. Somerespondents who score 0 may make one statement about how a strategy is related to a concept, but the statement istypically about the usefulness of manipulatives and does not go beyond that usefulness.

1. Responses scored 1 indicate a beginning view of mathematics as an interrelated web of concepts and procedures.Respondents describe connections between mathematical concepts and the strategies to justify the order in whichthey would have students share strategies. Some may even mention two connections between strategies, butbecause they want students to share only three strategies, the web that they have created is considered sparse.

2. Responses scored 2 indicate a view of mathematics as an interrelated web of concepts and procedures withdescriptions of connections between strategies. Because these respondents want to share at least four strategies,the view of mathematics as a web is more expansive than the view of someone who scores 1.

B1-S3.3

Scoring Summary


0

Shares 1 or 2 strategies.

Shares 3 strategies but explanation• indicates a progression from easiest to most difficult (with no strategy-to-strategy or concept-within-a-

strategy connections) OR• has no strategy-to-strategy connections OR• has at most 1 concept-within-a-strategy connection.

Shares 4 or 5 strategies but explanation• indicates a progression from easiest to most difficult (with no strategy-to-strategy or concept-within-a-

strategy connections) OR• has no explicit strategy-to-strategy connections OR• has only 1 concept-within-a-strategy connection (and no explicit strategy-to-strategy connections).

1

Shares 3 strategies AND explanation has• at least 1 explicit strategy-to-strategy connection OR• at least 2 concept-within-a-strategy connections.

Shares 4 or 5 strategies AND explanation has• 1 explicit strategy-to-strategy connection OR• at least 2 concept-within-a-strategy connections.

2 Shares 4 or 5 strategies AND at least 2 explicit strategy-to-strategy connections.

B1-S3.3

Comments on Scoring

A concept-within-a-strategy connection is a statement about one particular strategy; the statement indicates whylearning the strategy would be useful, conceptually (e.g., “I would share Sarah's method because she uses numbersense to solve it” or “Maria's method shows place value with the blocks.”).

A strategy-to-strategy connection is a statement about two or more strategies; the statement explicitly indicates that thestrategies are related in some way: “Maria's solution is the most basic and allows the use of manipulatives; Elliott'stakes away the manipulatives but uses the same method to solve the problem.” In this case, the respondent makes aconnection between Maria’s and Elliott’s strategies by mentioning that the approaches are the same although the toolsused are different.

Coders should treat a respondent’s selection of “No Preference” the same as “I don’t want to share any others.”

B1-S3.3

Examples

1

q3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Carlos Henry Elliott Sarah Maria I think the order may put the lesson in ascendingorder of difficulty

0 Although sharing five, therespondent simply describes aprogression from easiest tomost difficult.

2


Maria Henry Elliott Sarah Carlos Children should begin by using hands-onmanipulatives and work their way up to basicaddition

0 Respondent did not write aboutconnections either within astrategy or between strategies(in this example, there is acomment about manipulatives,but there are no commentsabout the affordances of themanipulatives).

3


Carlos Maria Sarah ‘No Preference NoPreference

Carlos' method was straightforward. Sarah's is agood way to logically show kids how to add.

0 One concept-within-a-strategyconnection: Sarah's strategylogically shows kids how toadd.

B1-S3.3

4


Maria Carlos Elliott Sarah Henry Maria’s answer was a fun way to learn theanswer, Carlos was more traditional to teach.Elliott’s was a another easy way to explain theanswer Sarah seems complicated but it was adifferent way to answer it . Henry, i would notprefer to teach that method. i would politelyexplain how it was confusing

0 Although all five strategieswere chosen and statementswere made about each one, therespondent does not discussmathematical concepts or howthe strategies might be relatedto one another.

5


Carlos Sarah Maria No Preference NoPreference

'I feel Carlos's way of approaching the solution ismost comprehensible; it would be less likely forthe child to get confused. Both Sarah and Mariahave alternate ways to find the answer andstudents should be exposed to alternate ways offinding one solution

0 In the explanation, thisrespondent mentions all threestrategies chosen but does notsay how they relate or whatconcept is important in anyone of them.

6


Carlos Henry Elliott Sarah Maria Maria's strategy applies the basic idea aboutaddition. Then Elliott's approach is the same butwithout all the blocks. Carlos' method is fast andeffective so I'd introduce it at the end after allunderstand the points being taught.

1 One explicit strategy-to-strategy connection: aconnection between Maria'sstrategy and Elliott's strategy.

B1-S3.3

7


Maria Elliott Carlos No Preference NoPreference

I think Maria should be first because it is a basicstep that explains how numbers look. It alsoallows the students to count which I think theyare more comfortable starting out with. Next Iwould show Elliott's method because he shows anunderstanding of place value. Last I wouldintroduce the standard algorithm.

1 Two concept-within-a-strategyconnections. In this case, therespondent describes conceptswithin Maria's and Elliott'sstrategies.

8


Maria Elliott Carlos No Preference NoPreference

Maria's method shows place value with theblocks, then Elliott's method is a good follow-upto Maria's, since it is the same thing without theblocks. Once kids have a good understanding ofplace value, they can transition to Carlos' methodsince they will know what the ones, tens, andhundreds place are.

1 Two explicit strategy-to-strategy connections (betweenthe strategies of Maria andElliott, and the strategies ofElliott and Carlos) BUT therespondent wanted only threepeople to share.

B1-S3.3

9


Carlos Henry Elliott Sarah Maria Maria's solution is the most basic and allows theuse of manipulatives. Elliott's takes away themanipulatives but uses the same method to solvethe problem. Henry's does the same, onlywithout written steps in the solution. Sarah'ssolution is a nice progression into comfort withnumbers, but Carlos' algorithm might be nice forstudents to know once they have anunderstanding and comfort with large numberslike these

2 Two explicit strategy-to-strategy connections AND therespondent wanted four ormore strategies shared. Theexplicit connections drawn arebetween strategies of Mariaand Elliott and between thoseof Elliott and Henry.

10


Maria Henry Sarah Elliott Carlos Maria's is visual and step by step addition.Henry’s is descriptive using words, and thenSarah’s because it gives the same idea, but withthe right answer. and then Elliott’s because it isthe same idea but in the head more, and lastCarlos because his is the end result

2 Two strategy-to-strategyconnections (between Maria'sand Sarah's strategies andbetween these two strategiesand Elliott's strategy), and fouror more strategies wereselected to be shared.

B1-S3.3

Training Exercises—Set 1

A


Maria Elliott Carlos no pref no prefA child has to begin somewhere and in order foranyone to learn you must begin small. Maria'sprocedure is a visual way to explain and lesscomplicated to learn for the first time.

B


Maria Henry Elliott Sarah Carlos First it would be good if the students can startout being able to visualise the problem they aredoing, like Maria demonstrates. After they canbegin to do that, they can do it in their minds, likeHenry, and then begin to write it on paper tovisualise it in a different sense, like Elliott. Thenthey can learn other strategies of learning, sothey can learn why and how the problem works.Finally they can learn the conventional method,that is easy to use, that Carlos demonstrates.

C


Carlos Maria Sarah Henry Elliott I think the simplest way was Carlos's strategy,Maria's shows you hands-on, how it works,Sarah's would practice estimation, Henry'sstrategy works and could be useful, and i don'tunderstand Elliott's strategy.

B1-S3.3

D


Maria Carlos Elliott Sarah HenryTo understand math, it is easiest for children touse hands-on manipulatives in the beginning andwork their way up to basic addition and carryingand so on.

E


Elliott Henry Carlos Maria no pref I chose Elliott first because he breaks down thenumbers into hundreds and tens, then when itcomes time to add it, it looks like a much simplerproblem. Henry's strategy would be a good followup because he also simplifies the problem byadding the tens with the tens the ones with onesand the hudreds with hundreds, it's actuallypretty much the same problem. Carlos's strategywould be next because it is essential that theylearn to "carry over" numbers. Carlos's strategyis the most commonly known/used. FinallyMaria's. Though Maria's strategy is my favorite, Iwaited to teach it because I think first thestudents must really understand addition to beable to really understand the manipulatives.

B1-S3.3

F


Maria Sarah Carlos no pref no prefI would start with the visual aid type lesson planin order to help the students picture what theywere actually doing. Then i would follow up withSarah's strategy. The estimation that is involvedgives the students a better understanding ofplace value. If they understand this key point,they will be able to add or subtract anything. Bythe students using a paper and a pen, they willget themselves closer to solving the problemquicker. The last lesson would be Carlos'because he does it the fastest and mostefficiently.

G


Elliott Sarah Carlos no preference no preferencethink Elliott’s and Sarah’s method should betaught first so that place value is understood.Then when the student does it Carlos’s way, theywill understand that algorithm better

B1-S3.3


H


Maria Elliott Henry Sarah Carlos 'I began with Maria, because I thought thatmanipulatives would offer a good base to start.Then I moved on to Elliott's way, because Ithought that adding by place value seemed likethe next logical step. Next, I chose Henry'smethod, because it is just emphasizing Elliott'sway, but is done mentally. Then, I pickedSarah's way, because they can now distinguishplace value since they have been using blocksand have been adding according to place value.Finally, I chose Carlos's way, because, by now,they are familiar with the fundamentals of addingmulti-digit numbers, and this would offer them aquick and simple standard to work with.

I


Maria Carlos Elliott 'I don't want toshare any

others

'I don't wantto share any

others

'First the children need to visualize the problembefore they go on to doing it with just numbersand Carlos' way is more direct than Elliott's or theother children.

J


Carlos Maria Elliott Henry Sarah 'start with standard, then show withmanipulatives, then Elliott's alternative method,then use henry's explanation and lastly show "thetrick" of estimation

B1-S3.3

K


Maria Carlos Elliott Sarah Henry 'I would start out with the students that used theeasy, plain to see steps and the ones thatrounded the numbers. Then I would go into amore advanced way of showing the numbers bybreaking them up in a word sentence.

L


Maria Henry Elliott Sarah Carlos I would start out with Maria, so the children canactually see what they are doing. Then I wouldmove on to Henry's because you begin to thinkabout place value and its importance. Then iwould move on to Elliott's because it is a lot likeHenry's but it is starting to be set up in the formof an equation. Then I would share Sarah'smethod because she uses number sense to solveit. Finally I would share Carlos' because I wouldwant to make sure that the children reallyunderstood what they were doing before theystarted using the algorithms to solve the problem.

.

M


Carlos Elliott Henry Maria Sarah It is better to start out teaching the simiplestways and then work up to all the shortcuts.

B1-S3.3

N


Carlos Maria Sarah Elliott Henry Carlos is the most basic and should be taught toall children, Maria's is helpful for those who needvisual support, Sarah's is helplful becauserounding is a farley simple thought process forchildren. Elliott's and and Henry's ways seem asif they would confuse children so I would probablyleave these out of a lesson and empasize on theother methods.

B1-S3.3


Z


Maria Elliott Carlos Sarah Henry the base ten blocks help students undderstandplace value better and physically shows grouping.elliott's method also explains place value. Carloscomes next because its the most commonmethod

Y


Maria Elliott Carlos no pref no pref I ranked my answers according to the stages inwhich I would like my students to learn multi-digitadding along with appreciating place values.

X


Maria Carlos Elliott I don't want toshare any

others

I don't wantto share any

others

first I would share marias strategy because, like isaid, its visual, and it would really help the kidsunderstand how addition works, how ten longsequal a flat and how ten singles equal a long,which explains the "carrying part" so then, theywould be ready for Carlos' s method, and wouldunderstand why it is that they carry a number.and finally, elliots way, in this method, there is nocarrying, and is very straighforward, the kidswould enjoy it. i wouldnt teach henry's becauseits a repetition of Elliott and i wouldnt teachsarahs because i think it could cause confusion.

B1-S3.3

W


Maria Elliott Carlos no preference no preferenceMaria's way is wonderful when first learningaddition. Elliott starts to show a goodunderstanding of place value, and Carlos showsthe most precise and widely used algorithm thatchildren should know how to use for the future.

V


Maria Elliott Henry Sarah Carlos I think it is best to start out with visual aids thatcan help children to understand place value rightoff the bat. Elliott's, Henry's, and Sarah'sthinking are all somewhat similiar, and are variousways of solving the problem in one's head.Carlos's thinking is best saved until the end,when the children already understand placevalue, and then the algorithm will be of moreimportance and value to them.

U


Maria Elliott Henry Carlos Sarah Marias would be the first because it is visual andeasy for the children to understand and see whatwas happening. Then elliott and henrys ideabecause it is also easy to understand and dealswith each place value. Carlos's would be nextbecause it is similar to elliott but is a faster wayof doing it. Then sarahs because they wouldhopefully have a better understanding of whatwas going on to understand her explanation.

B1-S3.3

T


Carlos Maria Elliott Henry Sarah Carlos and Maria’s were easier to understand, theothers made the problem too difficult.

B1-S3.3

Solutions for Training Exercises


A 0 One concepts-within-a-strategy connection (Maria)

B 2 Two strategy-to-strategy connections (Maria to Henry and Henry to Elliott) and shares all

C 1 At least two concepts-within-a-strategy connections

D 0 Progression from easiest to most difficult described

E 1One strategy-to-strategy connection (Elliott to Henry); also two concept-within-strategyconnections (Elliott and Henry). Either way, this response is scored 1.

F 1Two concept-within-a-strategy connections (Maria and Sarah). Maria's is a bit loose, but wegave the respondent credit for this statement as being conceptual because she mentioned"seeing what they were doing."

G 1Although response has two weak strategy-to-strategy connections (between Elliott's andSarah's strategies and between those strategies and Carlos's strategy), only three strategieswere chosen.

B1-S3.3


H 2Two concept-within-a-strategy connections; two strategy-to-strategy connections—(a) Henry’s isa mental version of Elliott’s way” and (b) the place-value aspects of Maria’s and Sarah’s ways.

I 0 One concept-within-strategy connection.

J 0 No mathematical connections mentioned.

K 1 A few, weak concept-within-strategy connections but no strategy-to-strategy connections.

L 1Three concept-within-a-strategy connections (For Maria, "See what they are doing”; for Henry,"place value"; for Sarah, "number sense"); makes explicit the strategy-to-strategy connectionbetween Henry's and Elliott's strategies.

M 0 Indicates a progression from easiest to most difficult.

N 1 Two concept-within-a-strategy connections—(a) Maria’s is visual; (b) Sarah’s involves rounding.

Z 1 Two concept-within-a-strategy connections (Maria’s and Elliott’s)

B1-S3.3


Y 0

Although this response sounds web-like, the response is too global to score higher than 0.The respondent does not mention how particular strategies are connected with one anotherand does not mention the concepts within the strategies (one could give the respondent creditfor the global connection of place value to strategies, but the response would still score 0).

X 1

One strategy-to-strategy connection (Maria to Carlos). Could not score 2 because therespondent shares only three. Also notice that the respondent makes a connection betweenHenry’s and Elliott’s strategies but does not value the connection because it was the reasonprovided for NOT sharing Henry's method; thus, this connection does not count toward the totalnumber of connections.

W 0One concept-within-a-strategy connection (Elliott’s). Reason for sharing Maria's does not countbecause the response is positive but does not mention concepts.

V 1One strategy-to-strategy connection (Elliott’s, Henry’s, and Sarah’s). The comment aboutMaria’s strategy is a concept-within-a-strategy connection.

U 2 Two strategy-to-strategy connections (Henry’s to Elliott’s, Elliott’s to Carlos’s) and shares all

T 0Progression from easiest to most difficult indicated (in fact, we are uncertain whether therespondent would share the last three strategies inasmuch as they were “too difficult”).

Scoren % n %

0 112 70% 84 53%1 44 28% 57 36%2 3 2% 18 11%Total 159 159

Pre Post


B2-S3

Rubric for Belief 2—Segments 3.4 and 3.5

Belief 2

One’s knowledge of how to apply mathematical procedures does not necessarily go with understanding of theunderlying concepts.


This item is set in the context of analyzing three students’ strategies for adding 149 plus 286. Carlos uses thestandard algorithm taught in the United States to solve the problem. Elliott uses an expanded algorithm: He firstadds the hundreds, then the tens, and then the ones, before totaling the three partial sums. Sarah uses acompensating strategy in that she adds 150 and 200, then 80 more, then 6 more, and then subtracts 1, to makeher sum equal to the sum of her original numbers, 149 +286. The respondents are asked whether Carlos coulduse and explain Elliott’s strategy (Item 3.4) and whether Carlos could use and explain Sarah’s strategy (Item 3.5).This item is presented to determine whether the respondent states that an individual may be able to perform aprocedure (such as the standard addition algorithm) but may not understand the underlying mathematical concepts(such as the place-value concepts one must understand to appropriately use the expanded addition algorithm andthe compensating strategy).

The highest score on this rubric goes to respondents who recognize that Carlos may not be able to use Elliott’s orSarah’s strategies, because Carlos may have only procedural knowledge, whereas the other two strategies requiresome understanding of the underlying place-value concepts. In contrast, the lowest score on this rubric goes torespondents who state that if Carlos can perform the standard algorithm, then he can likewise use the other twostrategies.

The scoring system for this rubric is different from the system for most rubrics in that scorers first score Items 3.4and 3.5 separately using the Scoring Summary Table and then use a second table to determine an overall scorethat is based on the pair of scores.

Carlos149 + 286

Written on paper

Henry149 + 286


Elliott149 + 286

Written on paper

Sarah149 + 286


MariaManipulatives

= 100Called a flat

= 10Called a long

= 1Called a single






Carlos Yes

No

Henry Yes

No

Elliott Yes

No

Sarah Yes

No

Maria Yes

No


330, 340, 350, 360, 370, 380, 390, 400, 410, 420'; then the singles, '421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435.'


First:

Second:

Third:

Fourth:

Fifth:





B2-S3

Overall Rubric Scores

0. Responses scored 0 indicate that children who are able to perform procedures correctly understand the concepts thatunderlie the procedures OR show no evidence that Carlos may have gaps in his understanding. Respondents suggest thatchildren who can solve addition problems using the standard algorithm can seamlessly solve addition problems usingmethods that require understanding of place value. However, some who score 0 may state that the student who uses thestandard algorithm would be unable to solve problems using the other methods because the other methods are invalid or tooconfusing.

1. Responses scored 1 also tend to indicate that children who are able to perform procedures correctly understand conceptsthat underlie the procedures. However, the respondent may also suggest that a child who can solve an addition problemusing the standard algorithm may be unable to solve the same problem using the alternative strategies because thealternatives are too different, instead of because they are more conceptual. They may state that Carlos could solve andexplain a problem using Elliott's method but that Sarah's is too different for Carlos to be able to use.

2. Responses scored 2 may indicate that one's knowledge of how to apply mathematical procedures does not necessarily gotogether with one's ability to solve problems in multiple ways. Respondents in this group tend to acknowledge that childrenwho can perform procedures may not be able to solve problems using alternative strategies, but at least one of theexplanations is poorly developed. A typical response scored 2 might indicate evidence of the belief in 3.4 but not in 3.5.

3. Responses scored 3 show respondents’ understanding that children who are able to perform procedures correctly may notunderstand underlying concepts. They include reasonably well-developed explanations to support this belief in both 3.4 and3.5. A 3 differs from a 2 because both responses tend to indicate that Carlos may have procedural, but not conceptual,understanding.

B2-S3

Scoring Summary

Use the following table to score Items 3.4 and 3.5 individually. Then use the Overall Rubric-Score Calculation table tocalculate the overall rubric score.

Score Rubric details for Responses 3.4 and 3.5, individually

0• Carlos can explain and solve using the alternative strategy OR• Carlos cannot explain the alternative strategy because the alternative strategy is too

confusing.

1• Carlos could NOT solve the problem because the strategies are too different OR• Conflictive (one part indicates that Carlos definitely understands place value and another part

indicates that he might not understand place value) OR• Carlos could NOT solve the problem; no rationale for the belief is given.

2 • Indicates evidence of the belief that Carlos may not understand underlying concepts, but theexplanations are not well developed

3 • Indicates evidence of the belief that Carlos may not understand underlying concepts, and theexplanations are well developed

Overall Rubric-Score Calculation

Score permutations for

Overall score (3.4, 3.5)

0 (0,0)

1 (0,1) (0,2) (1,0) (1,1) (2,0)

2 (0,3) (1,2) (1,3) (2,1) (2,2) (3,0) (3,1)

3 (2,3) (3,2) (3,3)

B2-S3

Examples

1 (Two respondents’ similar responses)3.4 3.5 3.4 Score 3.4 Comment 3.5 Score 3.5 Comment Overall Score

Yes, I think his level of understanding ishigh enough to be able to do Sarah'sstrategy.Yes, especially since Carlos is able tocarry without difficulty. He already hasa sense of place value.

Yes, because Elliott's strategy isdone in the same way as Carlos; butwith more stepsMaybe because Elliott's strategy isstraightforward

0 Carlos canexplainSarah'sstrategy.

0 Carlos can explainElliott's strategy.

0

23.4 3.5 3.4 Score 3.4 Comment 3.5 Score 3.5 Comment Overall Score

No, I don't think he could because atleast to me, Sarah's doesn't make anysense and I just don't think he would beable to understand the strategy

Probably not because Elliott's is aconfusing method of solving thatproblem and Carlos chose a verystraightforward method.

0 Sarah'sstrategymakes nosense (i.e., istooconfusing).

0 Elliott's strategyis confusing

0


No. Carlos' setup involves carrying andadding. Sarah's methods involvesrounding up and down and then havingto subtract the extras..

No. Elliott is working in the oppositedirection from Carlos.

1 Carlos couldnot solve theproblembecause thestrategies aretoo different.

1 Carlos could notsolve the problembecause thestrategies are toodifferent.

1


Yes because Sarah and Carlos showthey understand although Carlos mightnot understand and might just know howto carry a one.

Yes because Elliott and Carlos areboth dealing with placevalue…unless Carlos knows nothingabout it and he's just doin it by thebook.

1 Respondentprovidesconflictinginformationabout whatCarlosunderstands.

1 Respondentprovidesconflictinginformation aboutwhat Carlosunderstands.

1

B2-S3


I don’t think Carlos would have anunderstanding of what Sarah did.

He may not because he just addssingle digits in the algorithm used.

2 Respondentacknowledgesthat Carlosmay notunderstandSarah’sapproach butdoes not statewhy.

2 Respondentacknowledges thatCarlos may bethinking onlyabout single digitswhen he adds butdoes not mentionwhatunderstandingElliott’s methodwould require.

2


It would be hard to tell based only on hisprevious work. Without developingnumber sense I don't think he would.

He might not. Elliott uses moreadvanced steps than Carlos

2 Respondentacknowledgesthat Carlosmay notunderstandunderlyingconcepts butthe argumentsare not welldeveloped.

2 Difficult to tellwhether therespondent holdsthe belief. Therespondentdiscusses Elliott’shaving moreadvanced stepsthan Carlos, butdoes not discussexplicitly whatCarlos may or maynot understand.

2


It's hard to say because I'm not too sureof where Carlos' level of understandingis. I don't know if he is simply doing thealgorithm with no understanding.

Maybe, it would depend on Carlos'level of understanding. He mightonly know how to do the problem in arote fashion and not have muchunderstanding for what he is doing,so he wouldn't understand Elliott'sapproach.

3 Carlos maynotunderstand.

3 Carlos may knowonly how to do theprocedure withoutunderstanding theunderlying place-value concepts.

3

B2-S3


A3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

Yes I do because Carlos has alreadydeveloped that strategy throughlearning how to carry. I'm sure hecould probably help out in teachingSarah the shortcuts to addition andsubtraction.

Just the same I think thatCarlos is more mature in histhinking and he could have aninfluence on Elliotts additionand subtraction.

B3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

He might be able to make sense of it,but his being able to use thestandard algorithm does notnecessarily imply that heunderstands the more complexconcept of what he is adding,something that Sarah's methoddemonstrates.

Carlos would more likely to beable to explain Elliott's methodsince it is closer to thestandard algorithm method heuses than is Sarah's. It stillmight be difficult for him tounderstand this method sinceit requires a betterunderstanding of addition tocomplete.

C3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

No I don't think he could. Carlos'algorithim is neat and organizedwhere Sarah's strategy can beconfusing and messy. It might bedifficault for Carlos to follow the stepsaccurately

Yes i think he could. Elliotsmethod is similar to Carlos' inthat it is organized. I think hewould understand the problemwith little explination.

B2-S3

D3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

no. carlos shows carrying over whilesarah is rounded up and down andadding and subtracting

yes because both show placevalue although carlos mightnot understand place valueand might just know how tocarry over

E3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

No, they used different organizationskills, Elliott wrote his explanationand Sarah spoke her expalnation.

Yes, because they used thesame symbols andorganization to write out theproblem.

F3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

No he proably just understands thestandard way

No same answer

G3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

Probably, if he could grasp theunderstanding of not being so exactwith the problem.

No, i don't think he couldbecause at leas to me, itdoesn't make any sense and ijust don't think he would beable to understand thestrategy.

B2-S3


H3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'i dont know 'i dont know

I3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'I'm not sure if he would be able toexplain or make sense of Sarah'sstrategy. He could just be applyingbasic rules and using a standardalgorithm without understanding thereasoning behind it.

'Again, I am not sure if hewould be able to make senseof or explain Elliot's way,because he might not evenunderstand the reasoningbehind the algorithm he choseto use himself.

J3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'No, because she is using words,where Carlos did his work on a pieceof paper.

'yes, because Elliott did astrategy like Carlos, exceptthat his is a little different.

K3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'Carlos will might have a problem touse Sarah's srategy because she isrounding up and changing thenumbers while Carlos works withwaht hw has. This might show thatSarah thinks ahead.

'Carlos will might have aproblem with Elliott's strategybecause Carlos just add thenumber vertically from right toleft, while Elliott adds from leftto right and brakes thenumbers to make new ones.

B2-S3

L3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'I think that Carlos would make senseof Sarah's method, I do not think hewould use it becasue paper andpencil I think is faster, but I think hewould understand it.

'I think that carlos might havemore trouble with Elliotts waybecasue in the standardalgorithm, there really is noplace value. Carlos might justadd up 4 and 8 and forget thatthey are 40 and 80.

M3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'yes because it is a simple concept. 'not at all, he would probablybe confused by all thedifferent adding.

N3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'No, I don't think Carlos couldunderstand it because his methoddoes not allow much room formathematical thinking. I think thismethod is best when the childalready has an advanced level ofunderstanding addition, because ofall the short-cuts and abst

'I also don't think Carlos couldmake sense of Elliott'sstrategy because of the samereason for not understandingSarah's. He might see thenumbers as a disconnectedwhole (3 separate numbersfor "149"), and not as aconnected whole (one numberonly "one

O3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

No, I don't think Carlos couldunderstand it.

'I don’t think so.

B2-S3

P3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'Yes i think he can make sense of it,but i don't think that he would be ableto explain it because it is a littlecomplicating.

'I think that he could makesense of and explain thisstartegy because it is verybasic and easy to see whatwas done.

Q3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'Not really. Sarah's stratgey isconfusing and has too many steps.Carlos has a simple way of findingthe answer without changing thenumbers to make it easier.

'I think that Carlos might beable to work out Elliot'sstrategy. He seems tounderstand the additionprocess, but then of coarse ifhe has only been taught thisone way as the only way, hemight have trouble.

R3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'no. Carlo's way seems moreautomatic and I think he does it thisway because sombody told him thisis done this way. Sarah, on the otherhand seems like there was moreexploration to get the answer.

'Yes, the process is similar.

S3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

'I don't think so. It's possible, butwith the strategy Carlos uses, itdoesn't seem that he knows what thenumbers mean corresponding totheir place value. It just appears thathe follows a rule.

'Possibly more so than Sarah's,but he may have difficulty sincehe doesn't appear to knowwhat the place value of eachnumber means. Also, Elliotworks left to right and Carlosworks right to left. This mayprove confusing.

B2-S3


Z3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

possibly not..... he may have justbeen taught an algorithim

maybe

Y3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

No, Carlos's approach seemed todemonstrate that he thinks in a verylinear pattern, while Sarah's strategyseemed much more abstract. Thistype of strategy may prove difficultfor Carlos to make sense of.

I definitely think Carlos woulddo better with Elliott's strategythan Sarah's, based on thepattern of Elliott's work.

X3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

Carlos would have to put some extrathought into Sarah's strategy. Carlosmay know how it works, but Sarahcan show him more of why it does,through a new light. He may beconfused at first, but I think he couldgrasp it.

Carlos would probably be reallyconfused if he just looked atElliots problem without anyexplantion because he's notthinking about it in the sameway Elliot is. Elliot is breakingeach piece down.

W3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

maybe not because standardalgorithm tds to be so verbatum thatthey memorize the steps withoutknowing why.

maybe not again because elliotis good at plae values

B2-S3

V3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

It would be hard to tell based only onhis previuos work. I have no ideawhat his level of understanding inmathematics is, so it would be hardto say.

ditto.

U3.4 3.5

3.4Score

3.4 Comment3.5

Score3.5 Comment

OverallScore

I would hope so. Sometimes whenwe learn the standard algorhythm wedont nessessarily understand why itworks the way it does. I don't knowone can really make that judgementbecause we can't see whether Carlosunderstands why it works that way.

Again, I would hope so. I'mpretty sure that if he sawElliot's strategy he could makesense of it and explain tosomeone else what Elliot wasdoing.

B2-S3


Exercise3.4

Score3.5

ScoreOverallScore Comment

A 0 0 0 Both responses indicate that Carlos could solve both problems.

B 3 3 3Both responses indicate that Carlos may not understand the underlying place-valueconcepts even though he is able to execute a procedure.

C 0 0 03.4 Sarah's method is confusing and messy.3.5 Carlos has the understanding to execute Elliott's procedure.

D 1 1 13.4 The strategies are “too different.”3.5 Conflictive response.

E 1 0 13.4 Respondent mentions differences.3.5 Carlos should be able to understand Elliott's method.

F 2 2 23.4 Respondent seems to indicate that Carlos may not understand the underlying

concepts, but the response is not well developed.Response 3.5 is scored the same as Response 3.4.

G 0 0 0Carlos could solve the problem using Sarah's method. Respondent indicates thatElliott's way is too confusing (because the respondent herself was unable tounderstand the strategy).

B2-S3

Exercise3.4

Score3.5


H 0 0 0 Respondent does not know.

I 3 3 3In both responses the respondent provides a well-developed argument for whyCarlos may be unable to understand the strategies.

J 1 0 13.4 Strategies are different.3.5 Carlos will understand Elliott’s method.

K 1 1 1In both responses, the respondent suggests that Carlos would struggle becausethe strategies are too different.

L 0 3 23.4 Carlos will be able to solve the problem.3.5 Provides a well-developed rationale for why Carlos may not understandElliott’s approach.

M 0 0 03.4 Carlos will understand Sarah’s approach.3.5 Elliott’s approach is too confusing.

N 3 3 3 Both responses are extremely well developed.

O 1 1 1Both responses indicate that Carlos may be unable to use and explain the otherstrategies, but the respondent offers no rationale for this choice.

B2-S3

Exercise3.4

Score3.5


P 0 0 0 3.4 Alternative strategy is too complicated3.5 Easy to see what has been done Elliott’s strategy

Q 0 1 13.4 Alternative strategy is confusing3.5 Conflictive answer

R 3 0 23.4 Carlos's automatic, not as much thinking3.5 Yes, can do it.

S 3 3 3In both 3.4 and 3.5, respondent mentions Carlos's lack of place-value knowledge,which would lead to difficulty.

Z 2 1 2

3.4 Carlos may not understand the underlying concepts, but the response is notwell developed.3.5 No explanation provided, but because the response is not definitely positive, itis scored 1.

Y 1 0 1

3.4 Alternative strategies are too different; for Carlos to understand Sarah'sstrategy would be difficult.3.5 Appears to move toward indicating that Carlos could do a problem usingElliott's strategy.

X 0 1 1

3.4 Carlos could, with some extra thought, perform Sarah's strategy. Although therespondent mentions that Carlos might get confused, in the end, she states thatCarlos could use Sarah's strategy.3.5 Carlos could not solve the problem using Elliott's strategy because thestrategies are too different. The respondent , although again mentioning confusionin 3.5, mentions a difference between the two strategies.

B2-S3

Exercise3.4

Score3.5


W 3 2 3

3.4 Well-developed response that indicates that Carlos may not understand hisprocedure3.5 Tends to focus more on Elliott than on what Carlos may or may not understand(not well developed)

V 2 2 23.4 Too vague to be considered well developed3.5 Same as 3.4

U 3 0 23.4 Carlos may not understand place value.3.5 Carlos will be able to perform and understand Elliott's procedure.

Scoren % n %

0 96 60% 69 43%1 53 33% 44 28%2 7 4% 17 11%3 3 2% 29 18%Total 159 159

Pre Post

IMAP Results for Belief 2 Segment 3

B2–S4

Rubric for Belief 2—Segment 4

Belief 2

One’s knowledge of how to apply mathematical procedures does not necessarily go with understandingof the underlying concepts.


The focus of this rubric is on what the respondent states about Lexi’s work and whether Lexi understandswhat she is doing when she correctly executes the standard algorithm for subtraction. Respondents whonote that she may not understand what she is doing are providing evidence of the belief. Somerespondents note that Lexi’s way (the standard subtraction algorithm) is difficult to understand BUT, thisrealization is NOT equivalent to saying that when Lexi executes the algorithm correctly, she may notunderstand what she is doing. Coders, in initially using this rubric, often mistake comments about thedifficulty of understanding the algorithm for evidence of this belief.

Learning this rubric can be difficult, but once it is learned, coding is fairly easy and reliability should berelatively high. Contributing to the rubric’s difficulty is the fact that we did not ask directly, “Does Lexiunderstand what she is doing?” We avoided direct questions of this sort because we were assessingbeliefs rather than knowledge.

Pertinent responses for the belief are usually in Items 4.3, 4.7, or both, although evidence can be found inother parts of the segment. Read the entire response to ensure that all relevant aspects of the responseare considered.

4. Here are two approaches that children used to solve the problem 635 – 482.

Lexi 5613 5 – 4 8 2 1 5 3 Lexi says, "First I subtracted 2 from 5 and got 3. Then I couldn't subtract 8 from 3, so I borrowed. I crossed out the 6, wrote a 5, then put a 1 next to the 3. Now it's 13 minus 8 is 5. And then 5 minus 4 is 1, so my answer is 153."

Ariana

635 – 400 = 235235 – 30 = 205205 – 50 = 155155 – 2 = 153 482

Ariana says, "First I subtracted 400 and got 235. Then I subtracted 30 and got 205, and I subtracted 50 more and got 155. I needed to subtract 2 more and ended up with 153."

4.1 Does Lexi's reasoning make sense to you?

Yes No

4.2 Does Ariana's reasoning make sense to you?

Yes No

4.3. Which child (Lexi or Ariana) shows the greater mathematical understanding?

Lexi Ariana

Why?

4.4 Describe how Lexi would solve this item: 700 – 573.

4.5 Describe how Ariana would solve this item: 700 – 573.

Click on Submit when you are ready to submit your answers and continue.

Question 4 (continued)

Here are those two approaches again so that you can refer to them to finish this section.


Ariana

635 – 400 = 235235 – 30 = 205205 – 50 = 155155 – 2 = 153 482


For the remaining questions, assume that students have been exposed to both approaches.

4.6 Out of 10 students, how many do you think would choose Lexi's approach?

out of 10 students would choose Lexi's approach.

4.7 If 10 students used Lexi's approach, how many do you think would be successful in solving the problem 700 – 573?

out of 10 students would be successful.

Explain your thinking.

4.8 Out of 10 students, how many do you think would choose Ariana's approach?

out of 10 students would choose Ariana's approach.

4.9 If 10 students used Ariana's approach, how many do you think would be successful in solving the problem 700 – 573?

out of 10 students would be successful.


4.10 If you were the teacher, which approach would you prefer that your students use?

Please explain your choice.

B2–S4

Rubric Scores

0. Responses scored 0 indicate that Lexi’s correct execution of the procedure shows that she understands the procedure.Respondents may think that Ariana has the better understanding, but in their analyses of Lexi’s work, they give noindication that Lexi may not understand what she is doing.

1. Responses scored 1 include minimal evidence that Lexi may not understand. The respondents may provide a hint ofLexi’s not understanding when they discuss Ariana’s strategy. However, they directly state neither that Lexi doesunderstand nor that she may not understand. In contrasting Lexi’s and Ariana’s approaches, they indicate that Ariana hasunderstanding and that Lexi can get a correct answer. Without stating that Lexi may not understand, they imply, in the waythat they set up the contrast, that she may not.

2. Responses scored 2 show clear evidence of skepticism about Lexi’s understanding. Respondents state at somepoint—in commenting on which student has the better understanding, in considering whether students will be successfulusing Lexi’s approach, or in discussing which approach they would like their students to use—that Lexi may notunderstand.

B2–S4

Scoring Summary


0 • No mention of Lexi’s potential lack of understanding in either 4.3, 4.7, 4.9, or 4.10.• Think Lexi’s method is difficult but do not state whether it can be done without understanding.

1 • Indirect reference to algorithm as being not understood

2 • Specific mention of Lexi’s possibly not understanding what she is doing

Comments on Scoring

Pay attention to what respondent states about Lexi!

B2–S4

Examples

1q4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Lexi lexi's methodshows greaterunderstandingbecause it is amethod easy tolearn by andeasy topractice on,while araina onthe other handuses to manyrandomnumbers thanneeded.

10 10 all 10 studentswould besuccesfulbecause lexi'smethod is amuch betterway of learningandunderstanding.

0 0 she uses toomany randomnumbers tosolve theproblem.

Lexi's lexi's method isan easier wayof learning andteaching toother children.

0 No suggestion that Leximay not understand theprocedure she is using.


Ariana She understandthat numberscan be brokenup to get ananwser.

7 6 Most people doit this way,but alot of the timethey are wrongbecause theymake mistakesand do notcheck theirwork.

3 10 If they do it thisway they cancheck theirwork. aslo, theyreallyunderstand whatis going on andnot just doingthe process.

Both I think it isgood forstudents toknow how to aproblem morethan one way.That way theyhave choiceswhen they getstuck on aproblem.

1 An indirect reference toLexi’s not knowing “what isgoing on” in 4.9—“theyreally understand what isgoing on and not just doingthe process.”

B2–S4


Ariana

Lexi only seesthe digits asseparatenumbers andshe doesn’tunderstandwhat theyrepresent. Shesees 635 as 6 ,3, and 5instead of asthe wholenumber 635.Ariana on theother handunderstandswhat thenumbers mean.

5 6 Because in thisprocedure thereis a lot of roomfor error.

5 9 Because there isless room forerror since thementalsubtractionmade easier.

Ariana's

Because thereis less room forerror and itgives meaningto the problem.

2 The response includes adirect comment that Lexidoes not understand.

B2–S4


Aq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana They both showmathmaticalunderstandingbut Ariana isplaying with thenumbers tomake it moreunderstandableto herself. Leximay be justdoing a problemthe way shehas alwaysdone it and notunderstandingwhy.

7 5 Most kids haveproblems withborrowing andwith 2 zeros itwill be evenmore confusing.

3 8 It is a step bystep problem,but there is moremath to do andsome margin forerror.

Both I used to thinkthe way that Iknew was thebetter of thetwo witch islexis way, butboth are goodand some ofthe kids mayneed to do oneway or theother in order tounderstand.And that is thewhole point forthem tounderstand.

Bq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi I think that

Lexi’s methodis a little easierto understandthan Ariana’s.Ariana’s methodis too complexand I think sheis giving herselftoo muchunnecessarywork.

8 9 One may makeand error whensubtracting orthey mightforget to markdown a numberthat theyborrowed from.

2 7 Some studentsmay subtract thewrong amount orleave out a stepwhich would leadthemto anincorrectanswer.

Lexi's Lexi’s methodis a lot clearerandstraightforwardand there isless room forerror.

B2–S4

Cq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Lexi Lexi’s way isthe mostcommon wayand I think it’sprettyexplanitory andsimple.Ariana’s way isa little toocomplex andthere are a lotof numbers toremember but itis also easy topicture if writtendown.

9 7 Some childrenwill make anerror.

1 7 Some childrenwill make anerror.

Both Both ways arefairly simpleand easy tounderstand.Both wayscould be usedwith anynumbers so it’sgood to betaught differentways so youcan choosewhich way iseasiest for you.

Dq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana She could see

the quantitybeyondnumbers andwas subtractingin a horizontalmanner which isdifferent fromrote learning

7 7 I think thatwhen kids taketheir homeworkhome theirparents willknow how tosubtract thisway, so it isreinforced.

3 3 It seems to takelonger or moresteps, maybekids look forshortcuts,

Ariana

Shows betternumber sense.

B2–S4

Eq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Lexi Lexi’s approachseems mostcommon andmore simple ifyou know howto borrow.Ariana’sapproachseems moresimple to thosewho havedifficultyborrowing,although youcould getconfused thisapproach andmake an error.

8 8 A couple mightforget to borowfrom the 6 but Ifeel most wouldanswercorrectly.

7 7 Most wouldanswer correctlybut with so manysteps it givesmore opportunityfor errors.

Lexi’s becauselearning how toborrow insubtraction isimportant and ifeverybody did itAriana’s waythen they wouldavoid using theborrowingmethod. Also Ifeel more errorscan be madeusing Ariana’sway.

Fq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Lexi Lexi was ableto borrow fromthe hundredscolumn with outhaving to break80 down to 30& 50.

7 5 There arealways boundto be a fewhuman errorwhen kids aretrying to domath.

3 2 There are alwaysbound to be afew human errorwhen kids aretrying to domath.

Both I would wantthem to use theapproach thatthey were mostsuccessfulwith.

B2–S4


Gq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

'Ariana 'Even though Lexishows the usualway one maysolve thisproblem, I don'tthink she knowswhy she isborrowing. Arianadoes show thatshe is taking 4.82way from 635 in abroken downformat.

'8 '9 'I think thestudents wouldknow how tosolve thisproblem thestandard way, somost of themwould get it right.

'2 '9 'I think that if theychose Ariana'sway is becausethey understoodwhat she wasdoing, so thereforeit may be eaiserfor the studentswho chose to do itthis way easier. Ithink that about 9would get it rightbecause therewould be at leastone student

'Lexi's 'It is not onlyfaster I think tosolve, but I wouldexplain what theywere doing. Iwouldn't wantthem to learn itthis way becauseI said this is theway you do it, butunderstand whatand why there aresolving they waythey are.

Hq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

'Ariana 'arianas methodshows greatermathmaticalunderstandingbecause youhave to knowplace value andknow the the 6 in635 represents600 not just 6

'6 '8 'i allowed room for2 people to makean error when theare subtracting.mistakes happen!

'4 '3 'i think that 4 outof 10 wouldchoose thismethod. i thinkthat less peoplewould makemistakes whiledoing a problemthis way becauseyou have tounderstand placevalue so much. ibelieve that thenumbers areeasier to work withalso

'Ariana's

'i would wantthem to know howto do this onebecause i wantthem tounderstand placevalue when theyare working withmath.

B2–S4

Iq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

'Lexi 'Lexi understandshow to borrowwhen subtractingand Ariana doesnot seem tounderstand thisconcept and thismay be thereason she didthe problem theway she did.

'9 '10 'Because oncetaught Lexi's waythe problem is notso overwhelmyand complex forthe child. Thechild could thenbreak down theproblem usinglexi's method.

'1 '0 'Ariana's waymakes whatappears to be acomplex problemto a child evenmore complex anddrawn out.

'Lexi's 'My answer is thesame as above(4.7).

Jq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

'Ariana 'Ariana was ableto completelybreak down theproblemunerstandingevery aspect ofwhat shesubtracted.

'7 '8 'Some childrenget confused withborrowing.

'4 '8 'Some childrenwould have aproblem breakingdown a number orproblem.

'Both 'Lexi's way isultimately faster.However you areable to see everyaspect of theproblem Ariana'sway.

B2–S4

Kq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

'Ariana 'I really don'tknow which one,but I would haveto say Arianabecause it wouldbe hard to figureout what numbersto subtract fromthe number tomake it easier inthe end.

'8 '8 'I think 8 of 10students becauseyou have to knowthat somestudents aregoing to makesimple errorswhile doing thisproblem.

'2 '2 'I think that thestudents thatchoose Ariana'sway would be alittle bit ahead ofthe class and thewould want to findan easier way tofigure out theproblem.

'Both 'I would wanteveryone to learnLexi's way. Ifthey sould dothat, then I wouldteach themAriana's way. Ibelieve that youhave to know thebasic way beforeyou go and learnall the shortcuts.

Lq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

'Ariana 'adriana becauseshe knew whatws beingsubtracted soshe could changeit into somethingthat was easierfor her tounderstand. Lexiwas using andalgorithm so wedont know if sheunderstands whatshe is doing.

'8 '6 'because thestandardalgorithm beingused, easilycreates errorsthat for manychildren are hardto avoid.

'2 '2 'The methodleaved less marginfor error. Morechildren wil get itright, that is ifthey understnad it.

'Both 'I would like themto know morethan one way tosolve any oneproblem so itwould be nothingbut benificial frthem to knowmore than oneway.

B2–S4

Mq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

'Lexi 'because sheunderstands howto borrow andwhen she needsto barrow andwhy she need tobarrow. she alsogot the correctanswer

'7 '5 'i am not reallysure. i think theywould choose itbecause that iswhat they weretaught but alot ofkids have a veryhard timebarrowing acrosszeros.

'5 '7 'i just nowunderstood thisreasoning. thismight be easier forkids because they dont have tobarrow acrosszeros. i got mixedup with the 30 andthe 50 beingsubtracted and ithought theanswer was 4.82because it waswritten on thebottom. i didn'tlook at it hardenough.

'Lexi's 'because it will bemore helpful lateron in life whenthe get to hardermath. i wouldreally mind if theused ariana's aslong as theyunderstood theconcept. bothare really okay.

B2–S4


Zq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana While Lexi doesthe normalalgorithem, thealgorithem thatI use and doesshow a greatunderstanding,Arian show thatsheunderstandsnumbers, andplace values.

8 10 Unless youmake amathematicalerror, this is asure way toalways get theright answer.

3 6 While Ariansapproach showsa goodunderstanding inplace value, it isvery easy toforget a number,or to forget whatyou were doing.

Both I think Lexi's isa good way toget an accurateanswer, butArians is agood approachin learningnumber sense.

Yq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana Lexi does notunderstand theconcept ofborrowing. Sheis not reallyborrowing one'sbut tenth's andotherwise theone from thesix would makefour notthirteen.

7 5 I think that theborrowingconcept is hardfor kids tounderstand andthey canbecomeconfused.

3 3 I think that thisis can be a verysimple process,if it is explainedproperly. Thechildren thatknow how to doit this way aremore likely tomake lessmistakes.

Both It reallydepends on tehsudent and togive the optionof doing it bothways givesthem a choiceto feel mostcomfortable.

B2–S4

Xq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana Arianaunderstandshow to makethe numerswork for her

8 10 it is the easiestway to visulaizethe problem andwit fewer stepsthere is lessroom for error.

2 7 because of theadded stepserrors would bemade.

Both I think Ariana'sway is betterforunderstandingthe concepts ofmath and I amwondering if itis easier forchildren to domath from leftto write sincethey read thatway and movingfrom top tobottom isharder to learn.Once theconcepts ofmath areachieved it is abetter life skillto do mathlexi's way sinceit is shorter.

B2–S4

Wq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Lexi Ariana'sapproachdemonstratesher ability tomanipulatenumbers andbreak theproblem downto make iteasier for her tocompute, butLexi's approachseems todemonstrate,like Carlos's, asolidunderstandingof computationand arithmetic.

5 7 I think thatabout 7 of the10 would besuccessfulbecause,though theyshould all havethe concretemathematicalknowledge tocomplete theproblem as Lexidid, some mayhave difficultycompleting themulti-steparithmetic in theway that Lexidid.

5 9 Ariana'sapproach breaksthe problemdown into simplesubtraction,which should beeasier forstudents tocomplete.However, it islikely that themanner in whichAriana brokedown theproblem mayprove confusingto a selectnumber ofstudents.

Both Although Lexi'sstrategy provesthat she has asolidunderstandingof arithmeticandcomputation, Iwould also bepleased to seeAriana'sresponsebecause itdemonstrates aproblem-solvingapproach thatwould likelyprove useful inotherapplications.

B2–S4

Vq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana Lexi haslearned how tomanipulate themathematics ofthe subtraction,but Ariana hasexplained thereasoningbehind her workbetter.

5 7 Some childrenlose track whenthey aresubtracting andthey forget howmuch theyborrowed, andwhich columnthey borrowedfrom or whichcolumn theylent thenumbers too.They may notknow whetherthey borrowed100, or 10 whenthey borrowed1.

5 10 Ariana's methodkeeps a runningaccount of whathas beendeleted, and howmuch is leftbefore we deleteanymorenumbers.

Both Every childlearnsdifferently, andthe moreapproaches achild uses tosolve aproblem, thebetter theirunderstandingwill be of theentire conceptof numbers andtheir relation tosubtractionproblems.

Uq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana Ariana usedmany differentequations tofind the answerwhen Lexi usedthe easymethod.

10 7 I think thatLexi's approachwas what theywere taughthow to do butprobably a fewof the childrenwould makesimple errors.

0 0 Because theyprobably weren'ttaught how to dothis approachand there are somany steps tothis approachthat somewherealong the linethe might makea mistake.

Lexi's This is thestandardmethod ofsolvingsubtractionproblems. Thisis what will helpthem when theyare doingdivision.

B2–S4

Tq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Lexi She does it inthe way I wastaught and Ifind it to be awonderful greatway. Lexi willprobably getthe right answermore often thanAriana

6 4 Because a lotof errors cancome up withLexi's way.You mightborrow from thetens columnwrong and rightdown the rightnumber.

4 4 BecauseAriana way isa little easierand peoplecan roundnumbers towhat theycan subtract.

Ariana's If you useAriana's way youare most likelyto get to the atleast around theright answer.And if kids areusing their ownmethods thentheirunderstanding isgoing to begreater.

B2–S4



A 2“Lexi may be just doing a problem the way she has always done it and not understanding why.”This quote clearly indicates that respondent recognizes that Lexi may not understand what sheis doing.

B 0Respondent states that Lexi understands and seems to think that the algorithm is easy tounderstand.

C 0Respondent equates correct execution of the procedure with understanding: “Both ways arefairly simple and easy to understand.”

D 1Respondent writes of Ariana’s “subtracting in a horizontal manner which is different from rotelearning.” Without stating that Lexi’s way was learned by rote, she implies it by the previousstatement.

E 0Respondent, in equating borrowing with understanding, suggests that Lexi understands theprocedure she is using.

F 0 Respondent expresses no doubt that Lexi understands what she is doing.

G 2 Respondent expresses doubt that Lexi understands the algorithm.

B2–S4


H 1Respondent emphasizes that Ariana has good understanding of place value. She does notstate that Lexi does not understand place value, but her comment “know the 6 in 635represents 600 not just 6” indicates that she doubts Lexi’s understanding.

I 0 Respondent indicates that Lexi understands what she is doing.

J 1In writing that “Ariana was able to completely break down the problem” and that sheunderstands “every aspect of what she subtracted,” the respondent hints that Lexi may notunderstand every aspect of what she subtracted.

K 0The respondent, although noting that Ariana has the better understanding, gives no indicationthat she doubts Lexi’s understanding.

L 2The respondent refers directly to the possibility that Lexi may not understand what she isdoing.

M 0 The respondent suggests that Lexi understands what she is doing.

Z 0Even though the respondent chose Ariana as having the greater understanding, she statesthat Lexi shows “great understanding” and indicates that Lexi’s is a failsafe way to solveproblems.

Y 2Although this response has troubling aspects, the respondent does recognize that one cancorrectly execute a procedure without understanding it.

B2–S4


X 1The respondent appreciates Ariana’s approach and suggests that it will support children indeveloping concepts but never directly states that Lexi may not understand the approach sheis using.

W 0 The respondent states that Lexi understands what she is doing.

V 1

The respondent notes that Ariana “explained the reasoning behind her work better,” indicatingthat Lexi may not understand the reasoning behind hers. Although she points out why Lexi’sway is hard to understand, she does not indicate whether a child who executes the procedurecorrectly understands the underlying concepts.

U 0 The respondent does not indicate that Lexi may not understand.

T 0

The respondent states that Lexi’s approach is difficult to execute correctly but indicates thatcorrect execution of this procedure indicates understanding. Her last statement shows onlythat she thinks children who use their own methods understand better than others, not thatshe doubts Lexi’s understanding.

Scoren % n %

0 146 92% 83 52%1 4 3% 24 15%2 9 6% 52 33%Total 159 159

Pre Post


B2-S8


Belief 2

One’s knowledge of how to apply mathematical procedures does not necessarily go with understanding of the underlyingconcepts.


In Item 8.1, the respondents rank four fraction items, including “Understanding 1/5 X 1/8,” in terms of their relative difficultiesand explain their ranking. This rubric is based on Item 8.4, in which respondents are asked to respond to the question “Inquestion 8.1, we asked you to rank the difficulty of understanding 1/5 x 1/8. By understand, were you thinking of the ability toget the right answer? [Select yes or no]. Please explain your response.” In the latter item, the respondents discuss what theythink “to understand multiplication of fractions” means. We interpret their responses to the two items as providing confirmingor disconfirming evidence of this belief according to whether the respondents (a) consistently draw distinctions betweenunderstanding fraction multiplication and performing procedures (highest score) or (b) conclude that a student who canperform a procedure understands it (lowest score). Middle scores go to two types of respondents: (a) those who provideclear evidence of the belief in one item but disconfirming evidence of the belief in the other item (we consider that theserespondents hold the belief, albeit fragilely; that is, although they provide evidence of the belief in one item, it is context-dependent in that they will provide disconfirming evidence of the belief on another item); (b) those who provide vagueresponses to both items so that determining whether these respondents provide confirming or disconfirming evidence of thebelief is difficult.

Respondents who receive the lowest score may fail to recognize that understanding fraction multiplication requires more thanperforming a procedure. Such respondents may view mathematics as following rules without reason. This response differsfrom the response of one who states that anyone who can perform the procedure of fraction multiplication must necessarilyunderstand the underlying concepts of that procedure. In our scoring, however, we score both these types of responses 0.The reliability on scoring this rubric tends to be lower than on most rubrics, because the scorers must interpret the languageused and sometimes make inferences not required in other rubrics. The readers are advised to spend extra time discussingall examples and training exercises carefully, discussing aloud their rationales for the scores they have given.

8.1 Place the following four problems in rank order of difficulty for children to understand, and explainyour ordering (you may rank two or more items as being of equal difficulty). NOTE. Easiest = 1.

a) Understand

Please explain your rank:

b) Understand


c) Which fraction is larger, , or

are they same size?


d) Your friend Jake attends a birthday party at which five guests equally share a very large chocolate bar for dessert. You attend a different birthday party at which eight guests equally share a chocolate bar exactly the same size as the chocolate bar shared at the party Jake attended. Did Jake get more candy bar, did you get more candy bar, or did you and Jake each get the same amount of candy bar?


Consider the last two choices:

___ c) Which fraction is larger, , or are they same size?

___ d) Your friend Jake attends a birthday party at which five guests equally share a very large chocolate bar for dessert. You attend a different birthday party at which eight guests equally share a chocolate bar exactly the same size as the chocolate bar sharedat the party Jake attended. Did Jake get more candy bar, did you get more candy bar, or did you and Jake each get the same amount of candy bar?

8.2 Which of these two items did you rank as easier for children to understand?

Item c is easier than Item d.

Item d is easier than Item c.

Items c and d are equally difficult.

Please explain your answer.

8.3. In a previous question, you were asked to rank the difficulty of understanding By understand, were you thinking of the

ability to get the right answer?

Yes

No

8.4 On the last question you indicated that you were thinking about understanding as "getting the right answer." Were you also thinking of anything else? Please explain.

B2-S8

Rubric Scores

0. Responses scored 0 indicate that respondents think of only the procedure for multiplying fractions although they are asked torank the item on the difficulty of understanding 1/5 X 1/8. In their responses, these students do not distinguish betweenability to perform this procedure and understanding of the underlying concepts.

1. Responses scored 1 show some recognition that ability to perform procedures does not necessarily indicate understanding ofunderlying concepts, but this distinction is fragile. That is, the respondents may mention conceptual understanding in onepart of the response but mention only procedures in a different part.

2. Responses scored 2 indicate that understanding of the underlying concepts does not necessarily follow from ability toperform procedures. The respondents consistently distinguish between understanding concepts and performing procedures.

B2-S8

Scoring Summary

Read responses to Item 8.4 first; then read response to Item 8.1b.


0• Responses to 8.4 AND 8.1b refer to being able to perform the procedure [procedures, procedures].

• Either 8.4 or 8.1b refers only to being able to perform the procedure, AND the other response (8.4 or 8.1b)is too vague to interpret [procedures, vague].

1

• 8.4 AND 8.1b are both too vague to determine whether the respondent is referring to procedures orconcepts. [vague, vague].

• Either 8.4 or 8.1b indicates that the student is (a) thinking of conceptual understanding OR (b) making adistinction between conceptual and procedural understanding, AND the other response (8.1b or 8.4) refersonly to being able to perform the procedure [concepts, procedures].

2

• Either 8.1b or 8.4 is too vague to interpret, AND the other response indicates that the student is (a) thinkingof conceptual understanding OR (b) making a distinction between conceptual and procedural understanding[concepts, vague].

• Responses to both 8.1b AND 8.4 indicate that the student is (a) thinking of conceptual understanding OR(b) making a distinction between conceptual and procedural understanding [concepts, concepts].

Comments on Scoring

The words vague, concepts, and procedures in the brackets above are to help the scorer label each response separatelyand then find the rubric score. For example, if the response in 8.1b is vague and the response in 8.4 mentions onlyprocedures, then the overall response is labeled [procedures, vague] and, thus, is scored 0. If the response in 8.1bincludes either mention of conceptual understanding or a distinction between procedures and concepts and the responsein 8.4 is vague, then the overall response is labeled [concepts, vague] and, thus, is scored 2.

B2-S8

Examples

1q8.4 q8.1b Score Comment

if a child understands a problem then they getthe correct answer

simple multiplication 0 Respondent refers only to the procedure.[procedures, procedures].


When I was thinking of "understanding," I wasthinking of knowing how to do the problem, orusing the correct concepts.

This problem is easier for childrenthan addition. It's multiplicationand you multiply, no tricks

0 Response 8.4 is vague (for example, one could ask what therespondent means by using the correct concepts), and theresponse in 8.1b refers only to procedures.[vague, procedures].


I thought of understanding in being able to seewhat the problem was asking and then possibleways to solve the problem

I think this is easier than the firstone because a child can justmultiply straight across where inaddition there are more steps

1 Although Response 8.4 appears to refer to concepts, theresponse in 8.1b mentions only procedures.[concepts, procedures].


Getting the concept of multiplying one numbertimes another number

Multiplication just is a hardconcept to grasp.

1 Insufficient details given in either response to determine what therespondent means by the term concept [vague, vague].


No, I was thinking that the meaning of theproblem is what would be difficult to understand

I think this would give students ahard time. Understanding what1/5 of 1/8 is may be a hardconcept.

2 Response 8.4 seems to refer to concepts and 8.1b definitelyrefers to the concept.[concepts, concepts].

6q8.4 q8.1b explain Score Comment

I wasn't at all thinking about being able to do thealgorithm correctly. If that was all that wasinvolved, the multiplication would be easier thanthe addition problem. I was thinking of trulyunderstanding what 1/5 x 1/8 really means.

Knowing what multiplication offractions really means is tricky!

2 In 8.4 the respondent distinguishes between understandingprocedurally and conceptually and in 8.1b appears to talk aboutconcepts (the interpretation is confirmed by the response in 8.4).[concepts, vague].

B2-S8


Aq8.4 q8.1b explain Score Comment

also of the ease of themultiplication algorithm

multiplying straight acrossmight be easier

Bq8.4 q8.1b explain Score Comment

understanding whatexactly the problem isasking is different that"getting the right answer"b/c you can memorizehow to do the problemwithout knowing why youare doing it.

multiplication of fractionsis one of the hardestconcepts when workingwith fractions.

Cq8.4 q8.1b explain Score Comment

Yes I was thinking aboutunderstanding on how tochange the equationaround in order for a childto be able to compute theanswer.

The method used to solvethis problem is easier tounderstand and solve.

Dq8.4 q8.1b explain Score Comment

Knowing why we add, andhow to add is moreimportant than getting theright answer. Once achild fully knows why heis doing what he is doing,the correct answer shouldeventually follow.

No common denominatorneeds to be found, onlymultiplication of the twodenominators.

B2-S8

Eq8.4 q8.1b explain Score Comment

I was thinking that themeaning of the problem iswhat would be difficult tounderstand.

The meaning of thisproblem is what I thinkwould give students ahard time. Understandingwhat 1/5 of 1/8 is may bea hard concept.

Fq8.4 q8.1b explain Score Comment

to get the right answerthey also need to have aunderstanding of howthey got it.. so when Iwas thinking it was bothgetting the right answerand also understanding..better for them tounderstand the conceptthen to always get theright answer

I think that they will knowthat they need to multiplyacross the top and thenacross the bottom

B2-S8


Gq8.4 q8.1b explain Score Comment

I was thiking about thechild understanding howto take a fraction of afraction. How do you takesomething is is already afraction of a whole andthen take a fraction ofthat. This is somethingthat must be taughtvisually in order to thepoint across.

This one is most difficultbecause it is very hard toconsider a fraction of afraction. The simplealgorithm is verydeceptive when solvingthis problem because itmay be easier to get theanswer than adding thetwo but it may be moredifficult to conceptualize.

Hq8.4 q8.1b explain Score Comment

I was thinking of the stepsthe student would have toknow to solve the problemand see if the studentwould be capable ofsolving it.

I think it is easy tounderstand whenmultiplying fractions, onejust has to multiplyacross, unlike adding orsubtracting fractions.

Iq8.4 q8.1b explain Score Comment

understanding is simiplycomprehending aconcept, not coming upwith the correct answer.

multiplying fractionscreate confusion

Jq8.4 q8.1b explain Score Comment

Yes, also about what themultiplication of fractionmeans, how when youmultiply fractions, thenumber gets smaller.

This would be the easiestif the children knowmultiplication.

B2-S8

Kq8.4 q8.1b explain Score Comment

'About the concept ofmultiplying

multiplying fractionscreates confusion

Lq8.4 q8.1b explain Score Comment

'about the concept ofmultiplying

I believe that it ismore simple thanadding fractions.

Mq8.4 q8.1b explain Score Comment

'I actually thought aboutthe understanding of whythey got the answer but Idon't think that would bean easy concept to graspwhy.

'This problem is simple ifthe understandmultiplication.

Nq8.4 q8.1b explain Score Comment

'Conceptually, I think thatit is more important for thestudent to understandwhy he/she arrived at thatanswer. Even if theproblem was wrong, whenthe logic is understood,the student will ultimatelybe lead to the answer.

'Multiplying fractions is aneven more difficultconcept than addition andsubtraction. This problemuses more logic, and thestudent must know the"rules" for multiplying.

B2-S8


Zq8.4 q8.1b explain Score Comment

i was thinking aboutwhether the child wouldknow what the problem iseven asking. what 1/5 *1/8 means.

this problem i think wouldbe very difficult forchildren to solve, first ofall its a multiplicationproblem and im not sure ifyoung children thoroughlyunderstand multiplicationyet, and next its also infractions. two verydifficult things for children

Yq8.4 q8.1b explain Score Comment

In order to get the rightanswer the child will haveto know how to get theright answer. So i wasalso thinking about themethod that they wouldneed to go about.

I think this is easierbecause they can justmultiply the numeratorsand then thedenominators and get theanswer

Xq8.4 q8.1b explain Score Comment

understanding theconcept of multiplyingfractions.

multiplication seems hard

B2-S8

Wq8.4 q8.1b explain Score Comment

he might be able to getthe right answer but thatdoes not mean he knewhow he got there. hewouldnt be able to explainit on his own. he cant usewhat he knows to doanother one thats a littleharder

simple multiplication

Vq8.4 q8.1b explain Score Comment

yes, I was thinking if thechild would be able tounderstand why yousimply multiply acrossrather then finding acommon denominator

I believe this to be thesimplest because I thinkchildren will know thatwhen multiplying fractionsyou just multiply rightacross

Uq8.4 q8.1b explain Score Comment

I was thinking of how thechild may not understandwhich numbers they mustmultiply and how tomultiply them.

Multiplication of fractionswould be the hardestbecause I think the childwill be unsure of what iswhat.

Tq8.4 q8.1b explain Score Comment

Understanding is a bettergoal then getting theanswer right, this shouldbe the first priority

conceptual and visual

B2-S8



A 0 Both Responses 8.4 and 8.1b refer to the procedure [procedures, procedures].

B 2The respondent distinguishes between performing procedures and understandingunderlying concepts in both 8.4 and 8.1b. She does not mention procedures [concepts,concepts].

C 0The respondent mentions computing and the "method used to solve this problem" in 8.4and 8.1b but does not mention understanding of underlying concepts [procedures, vague].

D 1The respondent mentions the role of understanding why in 8.4 but mentions only theprocedure in 8.1b. Note that although the 8.4 response appears to refer to Item a ratherthan Item b, the focus is on understanding instead of procedures [concepts, procedures].

E 2Response 8.4 is not well developed (that is, determining what the respondent means by"meaning of the problem" is difficult), but in 8.1b the respondent explicitly states thatunderstanding the multiplication concept may be difficult [concepts, vague].

F 1

Response 8.4 appears to be reasonably well developed (the respondent seems todistinguish between the procedure and the concept), but in 8.1b the respondent mentionsthat multiplication means multiplying numerators and denominators [concepts,procedures].

G 2The respondent discusses concepts and distinguishes between concepts and proceduresin 8.4 and discusses understanding fraction multiplication conceptually in 8.1b [concepts,concepts].

B2-S8


H 0 Both Responses 8.4 and 8.1b refer to the procedure [procedures, procedures].

I 2The respondent distinguishes between performing procedures and understanding theunderlying concept in 8.4; Response 8.1b is vague [concepts, vague].

J 1The respondent mentions the role of understanding why in 8.4 but mentions only theprocedure in 8.1b [concepts, procedures].

K 1The responses are brief and unclear in whether referring to procedures or concepts [vague,vague].

L 0Although in 8.4 the respondent refers to the concept of multiplying, no details to explain theterm concept are given (thus it is vague); in 8.1b the procedure is mentioned [vague,procedures].

M 1The respondent mentions the role of understanding why in 8.4 but mentions only theprocedure in 8.1b [concepts, procedures].

N 2The respondent distinguishes between performing procedures and understanding theunderlying concept in 8.4; however, the response for 8.1b appears to be conflictive, so theresponse is categorized as vague [concepts, vague].

Z 2In 8.4 the respondent mentions the meaning of multiplication and in 8.1b mentions a globaldifficulty of understanding multiplication and fractions. We scored this response as a weak2 [concepts, vague].

B2-S8


Y 0The respondent mentions the method to get the right answer in 8.4, , seeming to refer tothe method of knowing what to multiply instead of to some conceptual method, then alsomentions the method of multiplying straight across in 8.1b [procedures, procedures].

X 1Response 8.1b is vague. The response overall is too brief for the intention of therespondent to be discerned [vague, vague].

W 1Response 8.4 is reasonably well developed, but only the ease of the procedure ismentioned in 8.1b [concepts, procedures].

V 1Although Response 8.4 includes a mention of why, the thrust of the argument is about"simply multiplying straight across"; for 8.1b the respondent also mentions only multiplyingstraight across [concepts, procedures].

U 0In 8.4 the respondent seems to be referring to the procedure, and 8.1b is confusing[procedures, vague].

T 2In this brief response, the respondent distinguishes between understanding and gettingcorrect answers; the response in 8.1b supports (albeit weakly) the response in 8.4[concepts, vague].

Scoren % n %

0 104 65% 78 49%1 45 28% 56 35%2 10 6% 25 16%Total 159 159

Pre Post


B3–S4

Rubric for Belief 3 — Segment 4

Belief 3

Understanding mathematical concepts is more powerful and more generative than remembering mathematicalprocedures.


This rubric is designed to assess whether the respondent believes that Ariana’s conceptual approach will bemore generative than Lexi’s algorithmic approach. By a generative approach we mean one that is used withgreat success and gives rise to future conceptual development. One indicator of this belief, in our interpretation,is the success rate that respondents predict for the two approaches. Respondents who expect a highersuccess rate for those using Ariana’s approach are seen as providing some evidence of the belief. Anotherindicator of this belief is which strategies the respondent would like to have children in his or her classroom use.Those who would like Ariana’s approach used indicate that they perceive this to be a generative strategy. Therespondent’s choice of the child having the greater understanding was used only to discriminate between thetop two scores.

Our reliability in coding for this rubric was high because coding is fairly objective. Occasionally a respondent’scomments contradict her response to another question. In these cases a more subjective analysis is in order,and the coder must consider which response to use in coding.



Ariana

635 – 400 = 235235 – 30 = 205205 – 50 = 155155 – 2 = 153 482



Yes No


Yes No


Lexi Ariana

Why?




Ariana

635 – 400 = 235235 – 30 = 205205 – 50 = 155155 – 2 = 153 482



4.6 Of 10 students, how many do you think would choose Lexi's approach?

of 10 students would choose Lexi's approach.


of 10 students would be successful.


4.8 Of 10 students, how many do you think would choose Ariana's approach?

of 10 students would choose Ariana's approach.




B3–S4

Rubric Scores

0. Responses scored 0 indicate that remembering the standard procedure Lexi uses will be more generative than using Ariana’sconceptual approach. Respondents state that children will be successful when using the procedure. They state that Ariana’sapproach is confusing and that children will be unsuccessful when using it. They want children to use the standard algorithm andavoid Ariana’s approach.

1. Two types of responses are scored 1. Respondents who state that children will be nearly equally successful in using Ariana’sand Lexi’s approaches but prefer that children use Lexi’s score 1. Although they acknowledge that students might use Ariana’swith success, our interpretation is they do not see her approach as powerful because they do not want their students to use it.Some see Ariana’s as the back-up strategy for students who cannot learn the standard algorithm, instead of seeing it as the morepowerful of the two approaches.

The other respondents who score 1 state that children using Lexi’s approach will be more successful than those using Ariana’s.Some of these respondents are impressed with Ariana’s strategy and state that it is quite difficult. They do not, however, indicatethat the conceptual approach is more generative and powerful than the standard procedure. They would allow the conceptualapproach to be used in their classrooms (along with the procedure), indicating that they see it as having some power orgenerativity.

2. Two types of responses are scored 2. Respondents who choose either Lexi or Ariana as having the better understanding andthink children using these strategies will be equally successful score 2. They see the conceptual approach as being equivalent inpower and generativity to the standard algorithm. They would like to have both approaches in their classroom, often stating thatone approach will suit some students whereas a different approach will suit other students.

The other response that is scored 2 is from those who choose Lexi as having the better understanding but state that children willbe more successful in using Ariana’s approach than in using Lexi’s. Thus, they suggest that the standard algorithm is morepowerful than the conceptual approach, but they also see the conceptual approach as being generative. They also would likeboth strategies to be used in their classroom, and some state that Ariana’s approach should precede Lexi’s.

3. Responses scored 3 indicate that Ariana has the better understanding and that children will be more successful using herapproach than using Lexi’s. Some respondents prefer that children use Ariana’s; others would allow both. These responsesshow strong evidence that the conceptual nature of Ariana’s approach will be powerful and generative. Some respondents notethat Ariana thinks about the whole quantity instead of thinking about the digits separately, as Lexi does.

B3–S4

Scoring Summary


0• 4.3 Either Lexi or Ariana

Compare 4.7 and 4.9: Lexi’s is seen as being easier than Ariana’s.4.10 Prefer to have children use Lexi’s approach

1

A. 4.3 Either Lexi or ArianaCompare 4.7 and 4.9: Lexi’s and Ariana’s approaches are seen as being about the same.4.10 Prefer Lexi’s

B. 4.3 Either Lexi or ArianaCompare 4.7 and 4.9: Lexi’s is seen as being easier than Ariana’s.4.10 Both

2

A. 4.3 Either Lexi or ArianaCompare 4.7 and 4.9: The two approaches are seen as being about (within 1) the same.4.10 Both

B. 4.3 LexiCompare 4.7 and 4.9: Ariana’s easier than Lexi’s4.10 Both

3• 4.3 Ariana

Compare 4.7 and 4.9: Ariana’s easier than Lexi’s4.10 Both or Ariana’s

Comments on Scoring

In answering Items 4.7 and 4.9, respondents sometimes fail to recognize that the number they select is meantto be out of 10 students and think that it is out of the number of children they predicted (in 4.6 and 4.8,respectively) would choose the strategy. Their error is usually obvious to the coder. Looking at the numbers ishelpful, but be careful in doing so.

B3–S4

Examples


Lexi “She put it inthe right formatand subtractedthe right way,borrowing fromthe rightnumbers.”

10 10 Because it iseasier to seeand understandand as long asthey understandthe process ofborrowing andsubtractingsingle digitsfrom right toleft.

0 0 because it istoocomplicatedwith too manydifferencenumbers.

Lexi’s it’s easier tounderstand andsee with twonumbers insteadof the 8 Arianaused.

0 Ariana’s approach isconsidered to besignificantly moredifficult than Lexi’sand not worthpursuing.

2 (Scored re Rubric Detail 1A)q4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi 'Because lexi

knows to borrowfrom either thetenths orhundredthsplace in order toget the correctanswer.

7 5 'I think 7 woulduse it becausemost wereprobably taughtthis way sothey wouldapproach it inthis way. 5would get itright because acouple of kidswould getmessed upwhen borrowingfrom thehundredths forthe tenths spot.

3 2 'If the childrenare comfortablewith thismethod, thereare less stepsto follow for theanswer. 2 Ithink would getit right becauseagain there areless steps tofollow and thesubtraction Ithink is easierto accomadatewhat they feelmostcomfortable

'Lexi's 'I would rather teachthis because when thestudents get to highergrade levels, thisapproach I believewould be used themost. If students inmy class had troublewith lexi's approachthen i would teachariana's as a back up.Also once the numbersin the problem getslarger than the moresteps it would take fordoing it and alsoscrath paperspace(ariana'sapproach).

1 Children’ssuccess rates onLexi’s andAriana’sstrategies aboutthe same. Weassumed thatrespondent feltthat 5 out of 7students would getLexi’s correct and2 out of 3 wouldget Ariana’scorrect.Respondentpreferred thatLexi’s strategy beused in herclassroom.

B3–S4

3 (Scored re Rubric Detail 1B)q4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana breaks theproblem downinto fourproblems,making it easierfor her tosubtract. Thisestimation ofthe problem andcombination ofthe numbersmeans sheknows how tobreak stuffdown.

7 9 I think this isthe easiestway to solve aproblem for achild and that9 out of 10 willget it right.

3 1 I think toomanystudentswould beconfused onhow to breakdown thenumber intohundreds,tens andones andmany wouldlose theirattention ortrain ofthought.

Both I think both ofthese approachesare useful tostudents. Lexi’sapproach issimple andstraightforward. Ialso think Ariana’sapproach to usingestimation ishelpful to studentand will beespecially helpfulin their later mathclasses.

1 Ariana’s approach isconsidered to besignificantly moredifficult than Lexi’s,but it is at leastbeing given “airtime” in theclassroom

4 (Scored re Rubric Detail 2A)q4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana 'She reasonedout why shetook each stepthat she did.Her method wasalso lesscomplicatedthan Lexi'swhich could getvery confusing.

7 7 'Because heapproach issimplesubtraction,its herdiscription ofit that isconfusing.

5 5 'Her methodlooks morecomplicatedand drawnout, when itreally issimple. But Ithinkstusentsautomaticallygo towardsthe simplerlookingproblems.

Both 'I think that itis importantthat thestudentsunderstandthe reasoningbehind both ofthe problems.They need toknow thatthere is morethan oine wayto solve thistype ofproblem andthe reasoningbehind each.

2 This respondentindicates thatchildren using Lexi’sand Ariana’s methodswill be equallysuccessful (7/7 vs.5/5). She would likeboth approaches usedin her classroom.

B3–S4

5 (Scored re Rubric Detail 2B)q4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Lexi Ariana’sapproachdemonstratesher ability tomanipulatenumbers andbreak theproblem down tomake it easierfor her tocompute, butLexi’s approachseems todemonstrate,like Chris’ asolidunderstandingof computationand arithmetic.

8 7 They shouldall have theconcretemathematicalknowledge tocomplete theproblem asLexi did,some mayhavedifficultycompletingthe multi-steparithmetic inthe way thatLexi did.

2 9 Ariana’sapproachbreaks theproblem downinto simplesubtractionwhich shouldbe easier forstudents tocomplete.However it islikely that themanner inwhich Arianabroke downthe problemmay prove tobe confusingto a selectnumber ofstudents.

Both Although Lexi’sstrategy provesthat she has asolidunderstanding ofarithmetic andcomputation, Iwould also bepleased to seeAriana’s responsebecause itdemonstrates aproblem-solvingapproach thatwould likely proveuseful in otherapplications.

2 Response includespraise for Ariana’smethod in the sensethat it is easier forchildren to executeand it will be given“air time” in theclassroom. Thisresponse does not getthe top score becauseLexi is considered tohave greaterunderstanding.


Ariana although both cancomplete theproblem withflying colors,Ariana cansubtract biggernumbers like 235– 30 and 205 –50, while Lexi’shardest problemwas subtracting 8from 13

5 5 Lexi’sstrategy ismore difficultin that youhave toborrow.Some maynotunderstand orforget.

5 7 if the childrentake thingsone at a timethey prove tobe successful.Ariana’sapproachtakes evenfamiliarnumbers thatare easier tosubtract.

Both I wouldn’t knowwhichprocedureworks best inthe mind of achild withoutexposing themto bothmethods.

3 Ariana’s approach isconsidered thesuperior approach inall regards. Arianais given higherregard for herunderstanding; herstrategy will be usedmore successfullyand will be given “airtime” in theclassroom.

B3–S4


Aq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana Lexi does notshow that shenows that sheis trying tosubtract 30from 80 or thatshe isborrowing 100.

3 3 I think manyof thestudentswould forgetto borrow the100 from the700 andreach ananswer of227.

7 7 I think if thechildren areable tosubtractwith tensthis is arelativelyeasyproblem.

Ariana I think thisdemonstratesa greatermathematicalunderstandingand is notsimplyprocedural. Iwould wantstudents to beable to explainthis type ofreasoningbefore movingon to Lexi'sstrategy.

Bq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana Lexi answeredthe questionstraight forwardand Arianaused a moredetailed thoughtprocess butcame up withthe answer.

9 8 Some kidshaveproblemstrying toborrow andsome justreverse thenumbers andsubtractanyway theycan.

1 5 more detailedthought

Lexi's easier tounderstand

B3–S4

Cq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi I believe that

Lexi has theknowledge onborrowing and ifAriana had thatknowledge thenthese two wouldbe in the sameboat. Arianastill needs totake the stepsneeded tofigure out theproblem butLexi haslearned of theshortcut andstill knows howto figure theproblem outAriana style butcan do it theshort way aswell.

5 10 I am not sureof Lexi's agegroup butboth of thesemethods aresuccessfulmethods andby choosingthis methodthesechildrenwould finishthe problem alot quicker.

5 10 Again I amnot sure ofthe age butthis methodis successfulas well andas long asthe studentsfollow thestepscorrectly theywill finish theproblem fine.

Both To learn I wouldprefer that theyuse Ariana'sapproach becauseit shows thechildren the maththat is involved inreal subtraction.Once they havethat method down Iwould like them tolearn to borrow. Iknow that when Ilearned to borrowmy teacher calledit stealing notborrowing. Iwasn't quite surewhere it came frombut after using themethod for sometime I felt morecomfortable andcould figure outwhen I was doingthe problem wrong.I feel that I was apretty averagestudent and thatmost childrenwould probablylearn the sameway as I did.

B3–S4

Dq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi she

understandsthe borrowingmethod

8 7 some haveproblems withborrowing andsome forgetto carry orsubraact theone theyhaveborrowed ,just simplemistakes

2 1 doing somuch workfor the oneproblem, mayget messy,allowing forerror

Both it is necessarythat they knowmore thatn oneapproach on howto solve a problem

Eq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi lexi borrows

and subtractscorrectly andgets the correctanswer quickly.ariana gets thecorrect answerbut there areway too manysteps involved

10 7 i think somekids willforget toborrow twice

0 1 ariana'smethod takesto manysteps andyou have tothink aboutrounding, etc

Lexi's lexi's format is theeasiet to see andunderstand. youcan borrow right onthe paper.

Fq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi i believe that

the simplier themethod theeasier it will beto understandsimilar types ofproblems

7 10 i think that if thestudentsunderstand theborrowingconcept theyshould besuccessful atanswering thisquestion

3 10 i think that if thestudentsunderstand thisconcept theyshould be ableto successfullyanswer thisquestion.

Lexi's This is the mostcommon approach.

B3–S4

Gq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana because Lexi onlysees the digits asseparate numbersand she doesn;tunderstand whatthey represent.She sees 635 as 6, 3, and 5 insteadof as the wholenumber 635.Ariana on the otherhand understandswhat the numbersmean.

5 6 because inthisprocedurethere is a lotof room forerror.

5 9 becausethere is lessroom for errorsince thementalsubtractionmade easier

Ariana's

Because there isless room forerror and it givesmeaning to theproblem

Hq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Aariana 'It seems thatAriana has abetterunderstandingbecause she isable tomanipulate thenumbers andsee what theyreally stand for.

6 7 'I think thatbecause youwould have toborrow all theway from thehundredsplace childrenmight make afew mistakesor not knowwhat to do.

4 8 'I think thatthis approachis easier for achild tounderstandand not asconfusing tothem as Lexi'sapproach.There is lessroom for errorsusing thisapproach.

Both 'I would teach mystudents Lexi'sapproach first andmake sure that theyhad a good grasp ofit. Then I wouldteach them Ariana'smethod. This methodis good when usinglarger numbers it isalso faster. Beforeteaching Ariana'smethod I would makesure that my studentshad a goodunderstanding ofLexi's method firstand what they areactually doing whenborrowing.

B3–S4

Iq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi 'ARIANA

PROBABLYDOESN'TUNDERSTANDPLACE VALUEBUT LEXIMIGHT

9 7 'SOMECHILDRENGETCONFUSEDWITHBORROWING

1 9 'IT ISBROKENDOWNSIMPLY ANDTHAT WILLPROBABLYCONFUSELESSSTUDENTS

Both 'I THINK THAT ACHILD SHOULDDO HATEVERWAY MAKESMORE SENSE TOTHEM AND WHICHTHEY ARE MOSTSUCCESSFUL AT

Jq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana 'Ariana has a

greatermathematicalunderstandingbecause shecan think thewhole problemnthrough, whileLexiunderstandshow tosubtract.

8 7 'Most of thechildrenwould set upthesubtractionproblem theway Lexi didbecause thatseems to bethe mostcommon waytaught. Afew of thestudents willprobally add73 to thezeros insteadof subtractingthem.

4 5 This way justseems like itwouldconfuss thechildren withtoo manysubtractionproblems.

'Lexi's

'I think for learningthe fundamentalsof math theyshould masterLexi's approachfirst, and learn tothink it out asAriana did whenthey are morecomfortable withLexi's method.That way they cancheck Ariana'sway with Lexi'sway to be surethey got it

B3–S4

Kq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentLexi 'She knows how

to subtract andborrow anddoesn't have toseparateeverything,although it ishelful inlearning

7 10 'Because it ispretty simpleand layedout. The onlyproblemwould be amistake insubtraction

3 8 'I think itwould be hardfor each newproblem forthe childrento break upthe numbersrandomly likethat

Lexi’s 'I want them toknow how andwhy to borrow andsubtract in aquick and easyway. Breakingthings up doesn'tteach themborrowing etc

Lq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana 'Because

Ariana showshow she got tothe answer bybreaking thenumber up andsubtracting itby each of thesmallernumbers,whereas Lexi'sappraoch justshows thebasics but doesnot really showhow she gotthere.

7 5 'I think thisappraoch issimpleenough tounderstand,but I thinkstudentsmight getconfused onthe borrowingfactor andalso maymakemistakes withtheirsubtraction.

3 7 'I think moststudentsmight not usethis onebecause itseems likemore work,but I thinkthey will beable tounderstandthe reasoningbehind it withthis and bemore likely toget it correct.

Both 'I would first likethem to learnAriana's becauseit gives them thereasoning why,but I would afterlike them to learnLexi's because itwill make doinglarger mathproblems easierand less timeconsuming.

B3–S4


Zq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score Comment

Ariana Lexi knows"how" but Idon't know thatshe knows"why."Ariana showssheunderstands bysubtracting thevalues of eachnumber at eachstep.

6 7 Because ifthey don'tknow whythey're doingit, theycould makelots ofmistakeswhenborrowing--orif they don'tborrow.

4 8 They couldreallyunderstandwhat they aredoing butcould stillmakemistakes.

Both I'd like if theystarted withAriana's methodwhichshows greaterunderstandingand only goes toLexi's when theyunderstand whythe procedureworks.

Yq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana She

understandshow to breakthe problemdown intosegments thatshe cancomprehend.Lexi just goesthrough theprocessthought habitrather thanactuallyknowing whyshe does whatshe does.

9 9 Her way wasthe way mostschoolsteach it andstudentswould justautomaticallystartborrowing.One of tenstudentsmight make amistake ifthey are notquitecomfortablewith thestrategy yet.

1 9 It seemsunorthodox,however isvery simpleto follow.Nine wouldget it rightbecause itmakes themost senseand a studentis less likelyto get lost.

Both Multiple methodswill give thechildren the bestunderstandingand since theyare effective, it isa great asset tobe open mindedand realize thatmany ways work.

B3–S4

Xq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana Lexi marches on as

if all the numbersshe weresubtracting werefrom the ones placevalue.

8 5 Subtractingfrom round100's can beconfusing.

2 5 I can't reallysay -- I don'tthink I'vefigured outthis approachwhen dealingwith 0's.

Both This is afrustratingquestion. Howcan I possiblyanswer if I don'tcompletlyunderstand bothapproaches?

Wq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana Though I am not

quite sure whyAriana wouldsubtract like shedid, I think that herway seemed moreadvanced. But shedoesn't explainherself very well Ihad to conclude onmy own where shewas getting thenumbers from. LIkeshe said, "Isubtracted 400 andgot 235. " If theequation wasn'tthere or if I just readher explaination, myquestion would be, "But from what areyou subtracting 400from? And would Ibe able to do thismethod if I had tosubtract from 0?"

9 6 Manystudents stillhave problemborrowingwhen there isa 0 involved.

1 1 On paper itlooks like alot moresteps thanLexi'smethod.

Lexi's Just because inLexi's approachI can teach mystudents aboutplace valuesthat eachnumber has. Atleast better thanAriana's method

B3–S4

Vq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana she was able to

break thealgorithemdown.

10 8 guess 0 3 it isconfusing ifthey are notuse to it

Both one is faster andone shows a goodunderstandind ofwhat is goingdown

Uq4.3 q4.3_comment q4.6 q4.7 q4.7_explain q4.8 q4.9 q4.9_explain q4.10 q4.10_explain Score CommentAriana although lexi's

way is moredefining,ariana's waymake moresense to me. ithink that sheunderstandsthe place valueand the varioussubtraction.she couldunderstand touse whatever isleft to subtractthe nextnumber. unlikelexi's way, ithink ariana ismoreunderstandable.lexi's way isshorter(seemslike it) but idon't think thatlexi understandwhat she wasdoing.

5 5 it is becausethe studentsdo notunderstandwhy they aredoing it andin so theycan makeeasymistakes

5 10 it is becausemanystudents willthink it isharder whenusing ariana'sway.everybodywill besuccessful indoing thisbecause theyhave less ofan error tomakebecause theproblem isvisuallypresented

both it is becauseariana can showthe student whythey are doingsubtraction andthat it is easier todo. the studentscan also learnlexi's waybecause her waymay seem shorterand moreconvenient forsome students. ithink that bothways can benefitthe studentsbecause eachstudents learningis different and ihave to put thatin considerationthat i cannot justuse one methodfor all thestudents.

B3–S4



A 3Ariana’s approach preferred in the classroom. Children will be more successful when using it,and it shows better understanding.

B 0Indicates that Ariana has the better understanding but that her strategy is more complicated andthat children will have trouble using it. Conceptual approach is not noted as being moregenerative for children.

C 2The two approaches are considered to be about the same in terms of ease of execution, andboth approaches will be allowed in the classroom.

D 1 Lexi’s is considered the easier approach, but Ariana’s is given “air time.”

E 0 Lexi’s approach is described as being superior in every way.

F 1The two approaches are considered equally difficult, but Lexi’s is the only approach that therespondent would like to have children using.

B3–S4


G 3Ariana’s way preferred in the classroom. Children will be more successful when using it, and itshows better understanding.

H 2

Success rates for the two strategies will be about the same (they are within 1); respondent isinterested in having students use both approaches. Her interest in beginning instruction withLexi’s strategy is troubling, but this aspect reflects a different belief (the timing of teachingconcepts and procedures).

I 2Type 2(B) response—Ariana’s approach is seen as being easier than Lexi’s (90% success ratevs. 70%), and both strategies will be allowed in the classroom. Because Lexi is chosen as thestudent with the better understanding, the response is scored 2.

J 1

Could be scored 0 or 1(B). Going strictly by the 4.9 response, one would score it 0. However,the respondent states interest in having her students acquainted with Ariana’s approach; thus itcould be scored 1. The respondent provides some appreciation for the generativity of concepts,so we scored the response 1.

K 0Lexi’s approach is considered the better strategy throughout. The respondent expresses noappreciation for the conceptual nature of Ariana’s approach.

L 3

Ariana has better understanding. Children will be more successful when using Ariana’sapproach, and children will be encouraged to use both strategies in the classroom. All thewritten comments support the view that Ariana’s conceptual approach will be more generativethan the standard algorithm.

Z 3Ariana’s is considered the superior strategy throughout. The 7 (q4.7) and 8 (q4.9) are close insize, but the other comments indicate that the respondent thinks that the concepts will begenerative.

B3–S4


Y 2Appreciation shown for Ariana’s approach, but respondent thinks that children will be equallysuccessful using either method.

X 2

This response is difficult to code because the respondent is unsure whether she understandsAriana’s strategy. She seems to think that understanding is generative: She is willing to haveboth strategies shared and notices the limitations of Lexi’s. With little to go on with thisresponse, we gave her the benefit of the doubt.

W 0Lexi’s approach is considered easier. The conceptual nature of Ariana’s approach is notapparent to the respondent.

V 1We can go only by the numbers in this response; the written comments provide little information.The respondent seems to think that children will be more successful when using Lexi’s but iswilling to have Ariana’s used by her students.

U 3

Ariana’s is considered the superior strategy throughout, as an easy strategy that can beunderstood. The comment that “it is because the students do not understand why they are doingit . . . so they can make easy mistakes” is interpreted as strong evidence that the respondentthinks that understanding concepts is more generative than memorizing procedures.

Scoren % n %

0 57 36% 19 12%1 61 38% 40 25%2 33 21% 53 33%3 8 5% 47 30%Total 159 159

Pre Post


B3–S9

Rubric for Belief 3––Segment 9

Belief 3

Understanding mathematical concepts is more powerful and more generative than remembering mathematical procedures.


The focus of this rubric is on what kind of instruction the respondent suggests so that more children will be successfulwith division of fractions in the future. Respondents who emphasize the role of practice do not provide evidence thatthey believe conceptual understanding is more generative than memorizing procedures. Those who note the difficultyof remembering what is not well understood provide evidence that they believe in the generativity of conceptualunderstanding.

Coders should first examine responses to 9.5 and 9.6, the responses that provide most of the information related tothis rubric. Scanning the answers in 9.1–9.4 will help to determine whether the respondent emphasizes theimportance of understanding. Some make recommendations about what the teacher should have done in theirresponses to 9.1 and 9.4. Responses to 9.3 are important only for distinguishing between scores of 2 and 3.

Click to see the next interview segment. View Video (High Speed Connection)View Video (56K Modem Connection)

9.1 Please write your reaction to this videoclip. Did anything stand out for you?

9.2 What do you think the child understands about division of fractions?

9.3 Would you expect this child to be able to solve a similar problem on her own 3 days after this session took place?

Yes No

Explain your answer.

http://coe.sdsu.edu/survey/IIIa_hi.htm

http://coe.sdsu.edu/survey/IIIa_lo.htm

Click to watch another videoclip: View Video (High Speed Connection)View Video (56K Modem Connection)

9.4 Comment on what happened in this video clip. (NOTE. This interview was conducted 3 days after the previous lesson on division of fractions.)

9.5 How typical is this child? If 100 children had this experience, how many of them would be able to solve a similar problem 3 days later? Explain.

of 100 children could solve a similar problem later.

9.6 Provide suggestions about what the teacher might do so that more children would be able to solve a similar problem in the future.

http://coe.sdsu.edu/survey/IIIb_hi.htm

http://coe.sdsu.edu/survey/IIIb_lo.htm

B3–S9

Rubric Scores

0. Responses scored 0 indicate that the girl knows the steps and that such knowledge is adequate. The respondentsrecommend that she practice more. They may state that she does not understand, but they do not recommend that theteacher do something to promote understanding. They do not show evidence of thinking that concepts are moregenerative than procedures, because they do not suggest that the teacher should focus on the concepts rather than theprocedure. Their responses are focused on the role of repetition in helping children remember procedures and mayinclude suggestions about mnemonic aids like songs, acronyms, or chants.

1. Responses scored 1 indicate that this girl does not understand and needs an explanation AND more practice. (Weinterpret such statements as indicating that the respondents value both memorization and understanding.) Theseresponses do not indicate that understanding is more helpful than memorization for remembering mathematics.

2. Responses scored 2 indicate that the child does not understand but that the child will be able to solve the problem in afew days. We interpret this response as indicating that procedures can be easily learned and remembered. When theserespondents see that the child was unsuccessful, they recant and state that most children will not remember if they do notunderstand. They state that the teacher needs to act to promote understanding.

3. Responses scored 3 indicate that the child does not understand and that because of this lack of understanding, the childwill not be able to solve the same problem in a few days. They indicate that most children will not retain information theydo not understand and that the teacher must act to promote understanding. We interpret these responses as strongevidence of the belief that with conceptual understanding, children (not only some children) will be able to solve problemswithout having to memorize; throughout these responses, the importance of understanding is emphasized.

B3–S9

Scoring Summary


0

• The student was unsuccessful because she did not practice enough. Suggestions for the teacher centered on morepractice.9.2 May state that she does not understand9.3 Yes or No9.4 She forgot. She did not practice.9.5 Lots of children forget without practice.9.6 Practice more (may include suggestions about memorization aids [songs, acronyms, etc.]).

1

• The student was unsuccessful because she did not understand. Suggestions for the teacher include both practiceand explanation.9.2 May state that she does not understand9.3 Yes or No9.5 Most children will forget.9.6 Explain why AND practice.

2

• The student was unsuccessful because she did not understand. Suggestion for the teacher is to promoteunderstanding (no mention of memory tricks). Answers should be fairly well developed with emphasis onunderstanding. In the responses should be a reference to the notion that understanding concepts will help childrenremember them.9.1 May state that the child was not taught conceptually9.2 May state she does not understand9.3 Yes9.4 She did not understand (may be stated).9.5 Most children will forget if they do not understand.9.6 Promote understanding by doing something more than explaining the algorithm.

3

• The student was unsuccessful because she did not understand. Suggestion for the teacher is to promoteunderstanding (no mention of memory tricks). Answers should be fairly well developed with emphasis onunderstanding. In the responses should be a reference to the notion that understanding concepts will help childrenremember them.9.1 May state that the child was not taught conceptually9.2 May state that she does not understand9.3 No9.4 She did not understand (may be stated).9.5 Most children will forget if they do not understand.9.6 Promote understanding.

B3–S9

Examples

1q9.1 q9.2 q9.3_

choiceq9.3 q9.4 q9.5_

numq9.5 q9.6 Score Comment

i was surprised that shecould do one by herself

she doesn'treallyunderstand. theflippingpart, shedoesnt'know why

No she doesnt'reallyunderstandwhat is goingon

it was sad tosee herfrusterared

60 Probablyhalf andeven moreif they hadmorepractice

more practice 0 Because of theemphasis onpractice. Noindication giventhatunderstandingwill help thestudentremember.

2q9.1 q9.2 q9.3_



GET THAT POOR GIRL AKLEENEX! AT FIRSTSHEDIDN'T SEEM TOUNDERSTAND WHAT THETEACHER HAD DONEBUT AFTER GOINGOVER A FEW MORE ANDTHEN WORKINGTHROUGH ONEHERSELF, SHE SEEMEDTO HAVE GOTTEN THEHANG OF IT. IT MIGHTBE DIFFERENT,HOWEVER, IF THEPAPERS FROM THEPREVIOUS PROBLEMSWERE REMOVED FROMTHE TABLE, AS SHEMIGHT HAV E JUSTBEEN COPYING THESTEPS FROM THOSE,ORUSING THEM TOHELP HER REMEMBERWHAT TO DO.

I THINKSHE ISSTARTINGTOUNDERSTAND THESTEPS TODOINGDIVISIONOFFRACTIONS, BUT ITMIGHTTAKE A BITMOREPRACTICETOUNDERSTAND IT.

No SINCE THISWAS HERFIRSTEXPOSURETO DIVISIONOFFRACTIONS,SHE WOULDPROBABLYNEED AREVIEWFIRST INORDER TOREMEMBERTHE STEPS.

THESTUDENTWASUNABLE TOPERFORMDIVISIONOFFRACTIONSA FEWDAYSAFTER SHEWAS FIRSTINTRODUCED TO THEM.SHE TRIEDTO FIGUREOUT THESTEPS BUTWASUNABLE TODO SO.

5 THEREWOULDPROBABLYBE A FEWVERYBRIGHTCHILDREN,OR SOMEWHO HADPRACTICEDTHIS TYPEOFPROBLEMAFTERINTRODUCEDTO IT, BUTMOSTWOULD NOTBE ABLE TOSOLVE ONEWITHOUT AREVIEW OFTHE STEPS.

A HOMEWORKGOING OVER THESTEPS OF THEEQUATION WOULDHELP WITH AREVIEW AFTERTHE INITIALINTRODUCTION. INADDITION, A BRIEFREVIEW A FEWDAYS LATERWOULD HELP TOREINFORCE THEIRLEARNING OF IT,AS WELL AS AREVIEWIMMEDIATELYPRECEDING THEWORKING OF APROBLEM.

0 Because ofemphasis onpractice. Noindication giventhatunderstandingwill help thestudentremember.

B3–S9

3q9.1 q9.2 q9.3_



i think that the child inthe clip with practice canlearn the method ofdividing and multiplyingfractions.

nothingquite yet,maybeplacing aone under awholenumber .

Yes yes, if shepracticedthe methodshe learnedfrom herteacher

she struggledonrememberingwhat to dowith thefraction asfar as what todo with theone and thethree.

40 i would say thatless than 50percent couldsolve this a fewdays later. itwould takepracticebecause it isdealing withfractions ratherthan just wholenumbers.

try to explainmore thoroughlyon what to do andwhy the divisionsign turns into amultiplication signwhen putting aone under thewhole number andallow the childrento practice more.

1 Emphasis is onpractice and anexplanation ofwhy theprocedureworks.

4q9.1 q9.2 q9.3_



I was very surprised tosee that after seeingthree examples and onlydoing one problem on herown, she remembered allthe steps to solving theproblem.

I think thechild onlyunderstandswhat theteacher hasshown herto do. Sheprobablydoesn'tunderstandwhy.

Yes I think thechild wouldbe able tosolve asimilarproblem in afew days,as long asshe hadsome typeof homeworkinvolvingthese typesof problemsor sometype ofreview.

The childcould notremember thesteps tofiguring outthe answer tothe problem.She wasobviouslyonly shownthe few daysbefore withnoexplanationof why andnot given anymore practiceat this typeof problem.

5 because notmany childrenwill rememberthose steps tosolving theproblem withoutunderstandingwhy they had todo those certainsteps.

If the teacherasked the childmore questionsand had giventhe childexplanation asto why thosesteps tookplace, I thinkmore childrenwould remembermore.

2 Argument forpractice andexplanation ofwhy seemsbalanced, but thescales are tippedtowardunderstandingwhen therespondentargues that,because of lackof understanding,only 5 (of 100)students willremember how todo the problem.Also, memorytricks are notmentioned.

B3–S9

5q9.1 q9.2 q9.3_



confusing to child, boring,meaningless

nothing No she wasvery good atmemorizingthealgorithm inthe shortperiod oftime, but itwill not staywith her.

painful towatch. Childhad nounderstandingof what wasbeing asked.Child took along time towrite the 6because sheknew from thebeginning thatshe didn'tknow how todo theproblem.

5 nounderstandingof what thedivision reallymeant

I don't know howyou showdivision withmanipulatives(yet) but youwould have tostart that way.The childrenwould have tohave manyexperiences tounderstand theconcept beforeeven showingthem thealgorithm.

3 Emphasis is thatmemorization is notgenerative andconceptualunderstanding isgenerative.

B3–S9

Training Exercises––Set 1

Aq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

Thestudentjustcopiedwhat theteacherdid.

sheunderstandsthearithmeticbut not theentireconcept.

Yes becauseshe willcopy whatshe did.

Shecouldn'tremember.

5 Very few becausethey don'tunderstand whatthe fractiondivided by awhole number is.

Explain howmany timesdoes 1/3 gointo 4. Andthey thenhave a littlestory behindtheirunderstanding.

Bq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

the childwas veryclear thatsheunderstood how todo theproblem.theteacherdid a fewproblemsand thenas long astheproblemfollowed tosameformat,the childwas ableto do theproblem

sheunderstandsthe order onhow to solvethe problem.

Yes the childseemed tohave aprettygoodunderstanding oftheproblemsand didnot haveto askanyquestions

i wassurprisedthat thechild forgothow to dothe problembut it nowmakessense tome that ifshe hadneverlearned thisbefore andthen wasgiven a fewday breakon how todo theproblemthat shemay beconfused

5 i think that she ispretty normal toforget the problemafter a few daybreak

go over theproblemsagain andagain for thechild andexplain whyyou flip thesecondfraction andwhy youchangedivision tomultiplication

B3–S9

Cq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

shelearnedquickly

the you flipflop thefractions

Yes sheseems tograsp itwellenough

she forgot! 5 she's very typicalI would haveforgotten too

practice, youcan't justteachsomething oneday and thengo back to itlater, they willforget

Dq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

The childdid well onworkingtheproblemalone.But onething thatstood outto me washow welldid sereallyunderstand thewhole'flippingover andmultiplying' thing.

Sheunderstandshow to gothrough themethod andget the rightanswer.

No Withoutpracticeno. Shehas onlyworked afewproblemsand she isgoing bymethod,not byunderstanding.

The childdid notremembertheprocessafter a fewdays whichshows justwhat Ithought,she has nounderstanding of theprocessjust ashortmemoryhow tocopy theteacer'sexamples.

5 This child is verytypical, with nounderstandingbehind a processvery few would beable to remembera few days later

Work on theunderstandingof fractions,what divisionof fractions isall about.Also work onthe reciprocalof frations,give exampleswhy therecipricalmultipliedworks thesame as theoriginaldivided.

B3–S9

Eq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

'Theconceptwasobviouslynew to thechild. Shedid it verywell forher firsttimelearningdivision offractions.

'I think thatsheunderstandsthe steps todividefractions.

No 'I thinkthat itwouldtake a fewmore daysofpracticefor her torememberit. Itusuallytakesmore thanone try topermanentlyrememberhow to dosomethingnew. Herbrainneeds abreak toabsorbthematerial,then try itagain in afew days.Repeating

'The childcould notrememberhow tosolve theproblem.Althoughtheconceptwasintroduced,it was notallowed tosink in thenberetaught.

10 'I think that mostchildren have tobe taughtsomething a few -several times onseperateoccasions toremembersomething totallynew.

'Just keepteaching it. Aday or twoafter the firstteching, theteacher couldhave done asampleproblem on theboard askingthe studentsfor the steps.As a group,students mayeachrememberdifferentportions of thesolution. Aftera groupproblem ortwo, thechildren couldbe asked tocomplete oneon their own.

B3–S9

Fq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

The childunderstood how touse thealgorithmto solvetheproblem,but shealmostseemedlike arobot.She usedthe exactsamewords andsteps asherteacher.She didn'tneed tothink atall, and Idon't thinksheunderstood whatshe wasdoing.

Not much. Ithink she wasjust repeatingwhat herteacher hadshown her,without anyunderstanding of divisionof fractionsor why shewas doingwhat she wasdoing.

No She mayremembersome steps ofthe formula,but until sheclearlyunderstandswhy she isdoing whatshe is doing,she won't beable to solvesimiliarproblems,especially ifthey are just alittle bitdifferent thanthe ones thatshe had beentaught tosolve with thealgorithm.That alwaysthrows a childoff track,when theproblem theyneed to solveis differentthan the onesthey learnedhow to solvewith thealgorithm,unless theyclearlyunderstandmathematicallywhy they aredoing whatthey aredoing.

That was what Ithought wasgoing tohappen. Thechild had beenable to repeatwhat theteacher haddone the daybefore, withoutanyunderstanding,like a robot.But when shehad to comeback the nextday, she didn'tknow what todo, or why todo it. It hadjust beennumbers and aformula to her,notunderstanding

20 I thinkthat somechildrenwouldrememberwhat todo, justbecausethey mayhave agoodmemory.Othersmay justtake agoodguess andbe able tosolve it.

I think the teacherwould need to usesome visual aidsand drawings to getthe children tobetter understand it.Even when I was achild, before I tookmy Math 210 class,I used to makedrawings to try tofigure out how tosolve mathproblems. That waswhat worked bestfor me, and wouldprobably work bestfor at least someother children. Shecould use real lifeexamples, but notword problemswritten down onpaper yet, and havethem solve thosereal life problems.Then the teacherneeds to relate thereal life problems tothe ones on thepaper. If she canclearly get acrossthe message as towhy the children aredoing what they'redoing, I think thatthey wouldunderstand and beable to solve similiarproblems in thefuture.

B3–S9


Gq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

the mainthing thatstood outwaas thesniffles. ithink shehad arunnyknows...uknow howkids are

the childknows how toget theanswer in afractiondivisionproblem. shedoesntunderstandwhat she isdoing or whyshe is doingit. she onlyflips andmultipilesbecausethats whatshe wastaught to do.she probablydoesnt evenknow what itmeans to'divide'.

No shedoesnthave agoodenoughunderstanding...notenoughpracticeeither

she couldntdo thesame typeof problembecauseshe neverreallyunderstoodit in thefirst place.she onlyknew howto do itwhile it wasfresh in hermind andhadexamplesin front ofher. shecouldntreasonthrough itbecausethe wayshe learnedhow to do itwas justremembering steps. ormaybe shejus hadother stuffon hermind...likewhats shegonna doat recess.

5 this child is rathertypical. mostchildren wouldforget how to dothe problemunless they get alot of practice outof school. andonly the reallysmart kids wouldremember.

teach themthe reasoningbehind theproblem...howdid they arriveto the answerand what doeseach stepmean? givethem morepractice.

B3–S9

Hq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

the childmimicedwhat theinstructorshowedher

rotememorizationof equatoinset-up

No it was notaconcreteunderstanding, justshort termmemoriztion

predictable 3 maybe a fewbrillant children,who mayberemember well bythat method orhave beenexposed todivisoin offractions at home.

make solvingthe problemsmore to reallife examples,use wordproblems

Iq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

The nevershowedthe girlwithdrawingswhat theyweredoing.The couldhavedrawn 4squares,thendevidedthem intothirds andcountedup thenumber ofthirdstherewere.

I think thathe childunderstandshow to devidethem usingnumbers, butshe has noidea why sheis doing it.She has nomeans ofapplying thedata toeverydaylive.

Yes The videosaid thatsherehearseda fewmoreproblems.Hopefullyshe willhavememorized theprocess.

She hadforgottenhow todevide withfractions. Iaslo don'trememberseeing atime frame,was thisthe nextday? Aweek later?A month?

10 Her match skillsare not out of thenorm.

think thatshowingillustrations ofwhat is beingdone orprovidingmanipulativeswould havehelped.

B3–S9

Jq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

The childlearnedveryquickly.

I think sheunderstandswhat she hasto do to solvethe problem,but no why.

Yes This childseemslike shewould beable tosolve asimilarproblem afew dayslater,becausesheunderstood it rightaway.

The childhad no ideahow to goabout tosolve theproblem.

If the studentshad some helpand were taughthow to correctlygo about solvingthis problem, thenyes, they wouldbe able to solve itin the future, butif once again,they were giving aproblem like it andnot taught how todo it prior to it,then once again,no, they wouldnot be able tosolve it.

The teachercan putproblems upon the boardand have thewhole classparticipate,then havethem doindividual workto see howeach studentis doing andhelp thosewho needmore help orput the ingroups so thatthe studentscan help eachother.

B3–S9

Kq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

THE GIRLDID AGOODJOB OFGETTINGTHERIGHTANSWERON THELASTPROBLEMBUT IMNOTSURE IFSHEACTUALLYUNDERSTOODWHATSHE WASDOING. ITHINKSHE WASJUSTCOPYINGWHATSHE HADDONEWITH THETEACHERBEFORE

SHEUNDERSTANDS HOW TOSOLVE THEPROBLEMSBUT NOTWHAT THEANSWERREFERS TOOR WHY SHESOLVED ITLIKE SHEDID

Yes I THINK IFSHEKEPTPRACTICING THEPROBLEMS THANYES SHEWOULDBE ABLETOSOLVEANOTHERONE

SHEFORGOTHOW TOSOLVETHEPROBLEM

10 I WOULD SAYTHIS CHILD ISPRETTYTYPICAL. IDOUBT MOSTCHILDRENWOULDREMEMBER HOWTO SOLVE THEPROBLEM

EXPLAIN TOTHE CHILDWHAT SHE ISDOING ANDWHY. IF THECHILD DOESNOTUNDERSTANDTHAT ITWOULD BEHARD FORTHEM TOREMEMBERHOW TOSOLVE THEPROBLEM

B3–S9

Lq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

The girlknew howto do theproblemafter theteachershowedher how todo it acouple oftimes.Then, thething thatstood outat me wasthat sheknew howto do byherself.

I think sheunderstandsthat she hasto flip thefraction andthen multiplyit. she alsounderstandsthat 4 is eualto 4 ones.

No because ithink it'sstill a newprocessthat shejustlearnedand 3days fromnow she'llprobablyforget.

She wasn'table to doa similardivision offractionproblemeven afterdoing 4 or5 problems3 daysago.

20 Children have asmaller memorycapacity. Theywon't rememberhow to do divisionwithout showingsome kind ofvisual aid to helpthem see theproblem.

i think theteacher shouldhave showedthe child howto do theproblem withblocks. Also,she shouldhave shownher a way tocheck heranswer.

Mq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

'I was alittle lostin thebeginning,but itlooks likethepracticehelped thestudent doa problemby herself.

'That youhave to flipthe secondfraction, addamultiplicationsmybol andthen mulitplyacross.

No 'what iftheproblemwasn't awholenumber inthebeinning.She wouldalwaysthink thattheanswerwas awhole.

'Obviouslythe childneeded tobereinforcedof theproblem alittle bitmore.

50 'Some child wouldremember theproblem, but mostlikely, I would betthat half wouldstruggle, becausethere is a lot ofsteps toremember.

'Maybe makeup an acroymnfor thechildren torememder thesteps.

B3–S9

Nq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

'not agoodmethod.

Nothing No 'she has noclue why she isdoing what sheis doing...there is no wayit could stick...she knows howto multiply....thats about it

'duh... ofcourse shecouldnt do it... iwanted to makea commentthough, that iforgot to makepreviously... ido not think theinterviewer did apoor job ofexplainging howto dividefractionsprocedurally... imean, as far asexplainginggoes, she did afantastic job...it was clear,concise,explained verywell... i dontthink the reasonthe child couldntsolve theproblem hadanthing to dowith that, it wasjust that simplymoving numbersaround... shejust had no ideawhy she wasdoing what

10 'of course there isthat 10 percent ofbrainiacs outthere who can doanything afteronly seeing itonce... i alwaysresented thattype...

'pictures!blocks!pizzas!anything!

B3–S9

Training Exercises––Set 3

Zq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

theteacherdid notexplainwhat shewas doingor whyshe wasdoing it.

not much,maybe thatyou have tousemultiplication

No her idaeof theconcept iswrong,hse wansttaughtwhy shewsasupposedto dosomethingshe wasjust toldto do.. oitprobalystayed inher shorttermmeomor.Maybershe mightrememberit but Ithink shewould runintoproblemssomewhare downthe lineeventually.

she hadforgotteneverythingbecauseshe had noreason toremember

10 soem would bebright enough, butthe majority wouldneed theunderstadnig tobe taught to themo see why theywere told toanswer theproblem the waybefore.

use visuals,explain whyshe chose toflip the secondnumberaround.

B3–S9

Yq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

The childlearnedtheprocess ofdividingfractions,but I don'tthink sheunderstood theconcept ofwhy youflip thesecondfraction.She justfollowedthealgorithm.

She canfollow thealgorithmused whendividing byfractions.

Yes If thechild canrememberthatalgorithm,sheshould beable tosolveanotherdivisionbyfractionsproblem.

The childforgotabout"flipping"thefraction.Perhaps ifshe wasgiven anexplanationof why wedo this,she wouldhaveremembered thealgorithm.

80 When learning anew concept inmath, I needed toget it repeated tome many timesbefore it wouldstick. Not only inone day, butconsecutively. Itis hard toremember how todo math when it isnot reapeated toyou, or reveiwedwith you beforeyou do theproblem

Explain to thechildren whyflipping thefractionsworks whendividing theminstead of justgiving themthe algorithm.

Xq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

Theteacherdid a goodjob ofexplainingherselfand thechild wsveryattentive.

I think sheunderstandsto divide youhave toreverse thesecondfraction, but Idon't thinksheundestandswhy she hasto.

Yes She had agoodunderstanding and ifshe triedagain in afew daysit maytake her alittle time,but I thinkshe'll getit.

She forgothow to dotheproblem.

5 The child is verytypical, andwithout a lot ofpractice they willall have problems.

Using anumber ofdifferentfractions insimiliarproblems.

B3–S9

Wq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

The littlegirl knowshow tosolve theproblemafter shepracticedthe stepsheneeded todo in orderto solvetheproblem.

I really don'tthink sheunderstandwhy you puta one underor why sheneeds to flipthe fractionover in orderto multiplyacross

Yes She wouldbe able tosolve it onher ownbecauseshealreadypracticedtheproblemenoughtimes

The childcould notfollowthorughbecauseshe did nothave anexample tofollow .

90 There is a fewchildren whocould notremmember howto solve thisbecause theymight not have agood memory

find anothermethod ofsolvingdivision.Maybe withpictures

Vq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

Sheremembered thesteps insolvingtheequation,andremembered whattheteacherhad toldher. Shewas ableto learnthem andapplythem tootherproblems.

I dont thinkshe quiteunderstandswhy you flipit, but that itssomethingyou do tosolve theproblem. Sheunderstandsthat you flipthe fractionand thenmultiply itstraightacross.

Yes I haveconfidence that thischildcouldsolvesimilatproblemsbecausesheknows thesteps andmethodsof how togo aboutsolvingsuch aproblem

She forgotthe stepsof how tosolve theproblem.

25 On furtherthought, if theprocess isntpracticed often, itcan be forgotten,especially sinceits just learned.

Explain whythe problemworks the wayit does.Possibilycreate asenario toexplain what isgoing on..or touse morevisual aids.Also, assignhomework onthe subject sothe studentscan continuelearning theprocess andmemorize it athome, andthen teach itmore in class.

B3–S9

Uq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5_num q9.5 q9.6 Score Comment

I wasextremelysurprisedthat theyoung girlwas ableto do theproblemon herown andget therightanswer aswell. Ireally didnot thinksheactuallyknew whyshe wasdoing whatshe wasdoing, thesteps, itseemed tome, werefrommemorizing thesteps fromthepreviousproblems.

I really don’tthink sheunderstandsa whole lot,she knowsthat you haveto do certainsteps and flipbut shedoesn’tunderstandwhy.

Yes Yes, butonlybased offhermemory ofhow toperformthe stepsand in theright ordertoo.

I wasn’tsurprisedat all thatshe wasn’table to dotheproblem, Ialready feltthat if shewas able todo theproblem, itwould havebeen frommemory.It’s hard toperformoperationsif you don’tknow whyyou’redoing them.

25 Without anexplanation ofwhy you’re doingthings, anyone,not just children,have a hard timeknowing how todo it the nexttime. The 25 thatI think may knowhow to do it,would be thechildren with goodmemories.

The teacherneeds toexplain whyshe is doingthe things shedoes, picturesmay help.

B3–S9



A 2Respondent mentions lack of understanding in 9.2 and 9.5. Her meaning for “give them alittle story behind their understanding” is unclear, but we assumed that she wanted thestudents to have some conceptual understanding.

B 1 This is a classic example of an “explain and practice” sort of response.

C 0Another classic; respondent suggests that practice will ensure success on this type ofproblem.

D 3

Respondent questions the child’s understanding of the concept from the start. In 9.3 shenotes that lack of understanding will interfere with the child’s ability to solve future problems,and she again mentions lack of understanding in 9.4 and 9.5. Finally, in 9.6, she notes theimportance of teaching for understanding. Clearly, throughout the responses, the emphasisis on understanding.

E 0 Respondent advocates more practice.

F 3 Respondent provides a strong and personal rationale for the importance of understanding.

G 1 Respondent wanted to provide the children with an explanation and have them practice.

B3–S9


H 3

Respondent saw limitations from the start. “No” in 9.3 and “it was not a concreteunderstanding, just short term memorization” in 9.4 indicate the belief that understandingconcepts will help children remember them. Her 9.6 response indicates a conceptualapproach to the topic.

I 2Respondent advocates teaching beyond the explanation of the algorithm but also believesthat the child will be able to answer future problems as a result of the instruction shown inthe first video.

J 0 More practice is what children need.

K 1

Respondent recommends practice and explanation. Although the comment that “if the childdoes not understand that, it would be hard for them to remember how to solve the problem”(in 9.6) is a favorable response, we concluded that the respondent was advocating nothingmore than an explanation of the algorithm.

L 3Respondent doubts that the child will be able to solve future problems and advocates aconceptual approach to teaching the content.

M 0The emphasis in this response is on memorizing an algorithm rather than onunderstanding the concepts.

N 3The emphasis throughout is on understanding, and the respondent differentiates betweenprocedural and conceptual understanding.

Z 3Respondent doubts that the child will remember. The suggestion for the use of visualsindicates a conceptual approach that will help children remember mathematical ideas.

B3–S9


Y 1Emphasis is on practice and explaining the algorithm (as opposed to understanding theconcepts).

X 0 The child needs more practice.

W 1

The 9.6 comment, “find another method of solving division. Maybe with pictures,” is arecommendation for an explanation that goes beyond explaining the algorithm. But therespondent also indicates that the instruction shown in the video was sufficient for thestudent to remember (only a few will not remember because of bad memories).

V 1 The emphasis is on practice and on explaining the algorithm.

U 2

The suggestion to explain with pictures why the algorithm works and the assumption thatonly students with good memories will remember the algorithm indicate a propensity towarda perspective that understanding mathematical concepts is more powerful thanmemorization.

Scoren % n %

0 91 57% 56 35%1 42 26% 32 20%2 9 6% 4 3%3 14 9% 67 42%Total 156 159

Pre Post


B4–S3.3

Rubric for Belief 4––Segment 3.3

Belief 4

If students learn mathematical concepts before they learn procedures, they are more likely to understand the procedures whenthey learn them. If they learn the procedures first, they are less likely ever to learn the concepts.


This item was designed to assess respondents’ beliefs about whether children should learn concepts first so that thechildren are better able to understand standard algorithms when they learn them. Respondents are placed in the role ofclassroom teachers and are asked to select an order in which they would focus on particular strategies during a unit onmultidigit addition. Respondents who choose to first teach the standard addition algorithm receive the lowest score,because they provide disconfirming evidence of this belief. Respondents who receive the highest score choose to teachthe standard addition algorithm fourth or fifth and describe the conceptual progression they would want their students toexperience prior to learning this procedure. These respondents explicitly mention that children should understandunderlying concepts (such as place value) before they learn standard algorithms.

Carlos149 + 286

Written on paper

Henry149 + 286


Elliott149 + 286

Written on paper

Sarah149 + 286


MariaManipulatives

= 100Called a flat

= 10Called a long

= 1Called a single






Carlos Yes

No

Henry Yes

No

Elliott Yes

No

Sarah Yes

No

Maria Yes

No


330, 340, 350, 360, 370, 380, 390, 400, 410, 420'; then the singles, '421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435.'


First:

Second:

Third:

Fourth:

Fifth:





B4–S3.3

Rubric Scores

0. Responses scored 0 indicate that children should learn standard algorithms before concepts. Respondents may selectthis order for many reasons, but often they state that learning the standard algorithm first will make learning the concept ata later date easier than if the students had learned the concepts first and then were taught the standard algorithms. Weinterpret the choice to share Carlos's strategy first (3.3) as exemplifying that belief.

1. Responses scored 1 still tend to indicate that children should learn standard algorithms before concepts, but therespondents may have an even greater focus on a progression that goes from what they consider the easiest strategy tothe most difficult. We consider this belief to be exemplified in Item 3.3 if respondents choose Carlos’s strategy second; theirexplanations often indicate their desire to order the strategies from easiest to most difficult (in particular they do NOTmention a desire to teach concepts and then standard algorithms).

2. Responses scored 2 indicate some value in teaching the standard algorithm after the students have been exposed to moreconceptual strategies. This aspect of the belief is evidenced, in our view, by the fact that the respondents choose to haveCarlos share 3rd, 4th, or 5th. However, their explanations do not indicate a desire to teach concepts before standardalgorithms. Some other respondents who score 2 do indicate a desire to teach concepts and then standard algorithms, butthey want Carlos to share 2nd. Having Carlos share 2nd (even with a reasonable concepts-to-procedures response)indicates a weak interest in having students learn mathematical concepts first, because learning just one of the moreconceptual strategies first may not provide a strong foundation for students to learn concepts before standard algorithms.

3. Responses scored 3 explicitly indicate an interest in having children learn concepts before standard algorithms. Therespondents may choose Carlos to share 4th or 5th, but they do not mention particular aspects of the conceptual progressionthey selected. Respondents may choose Carlos to share 3rd and indicate a desire to teach concepts before standardalgorithms, but having Carlos share 3rd (not 4th or 5th) indicates a moderate but not strong desire for students to learnconcepts well before they learn standard algorithms.

4. Responses scored 4 indicate that students should have a strong conceptual foundation before they learn standardalgorithms. We consider this belief to be indicated in item 3.3 by respondents who both want Carlos to share 4th or 5th andprovide details about the conceptual progression they would like their students to experience before they learn standardalgorithms. Quite often, a response of 4 includes an explicit statement that the students should understand, for example,place value before learning a standard algorithm.

B4–S3.3

Scoring Summary


0 • Chooses Carlos 1st

1 • Chooses Carlos 2nd AND the explanation does NOT indicate a desire to begin with concepts andthen teach standard algorithms.

2

A. Chooses Carlos 2nd AND the explanation indicates a desire to begin with concepts and then teachstandard algorithms OR

B. Chooses Carlos 3rd, 4th, or 5th AND the explanation does NOT indicate a desire to begin withconcepts and then teach standard algorithms

3

A. Chooses Carlos 3rd AND the explanation indicates a desire to begin with concepts and then teachstandard algorithms OR

B. Chooses Carlos 4th or 5th AND the explanation indicates a desire to begin with concepts and thenteach standard algorithms but explanation is not robust enough to merit a score of 4

4 • Chooses Carlos 4th or 5th AND the explanation indicates a desire to begin with concepts and thenteach standard algorithms AND includes particulars about the conceptual progression.

Comments on Scoring

For each ranking, if the same student is selected two times, use the first ranking for each person (for example, if aparticipant writes that Carlos should share second and fifth, for purposes of scoring, assume that Carlos shares second).

B4–S3.3

Examples

1q3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Carlos Henry Elliott Sarah Maria 0 No need to look atexplanation because Carlosis first.


Henry Carlos Elliott Sarah Maria I feel it shows a progression fromeasiest to most difficult

1 The explanation does notindicate a progression fromconcepts to procedures andCarlos is 2nd

3 (Scored re Rubric Detail 2A)q3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Carlos Elliott Sarah Henry I would share Maria's explanationwith the blocks so the childrencould see what they are addingthen I would use Carlos' so theywill learn how to do the standardalgorithm.

2 The respondent indicates adesire to begin withconcepts (with blocks sothey can see what they areadding) and then teachstandard algorithms.

4 (Scored re Rubric Detail 2B)q3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Henry Elliott Carlos Sarah Maria I went from what I think is thefoundation up to the mostcomplicated but most practical

2 The respondent does notindicate a desire to beginwith a progression fromconcepts to procedures.


Henry Elliott Sarah Maria Carlos I would definitely focus the most onSarah's or Henry's strategies

2 The respondent does notindicate a desire to beginwith a progression fromconcepts to procedures.

B4–S3.3

6 (Scored re Rubric Detail 3A)q3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Henry Elliott Carlos Sarah Maria I think it's important to understandplace value before you teachCarlos' method.

3 The respondent indicates adesire to begin with aprogression from conceptsto procedures.


Henry Elliott Sarah Carlos Maria I would want to work my way up tothe traditional algorithm makingsure that it is understood why wedo it that way.

3 The respondent indicates adesire to begin with aprogression from conceptsto procedures but does notinclude specifics about theconceptual progression.


Maria Henry Sarah Elliott Carlos First they should physically see itand see what they are doing, thenI think they should get a numbersense about it in the two differentforms like Henry and Sarah, then Ithink Elliottt's way is excellent toshow place value and then I wouldfinish up with the standardalgorithm.

4 The respondent indicates adesire to begin with aprogression from conceptsto procedures and includesspecifics about theconceptual progression(Henry and Sarah, Elliotttfor place value).

B4–S3.3


Aq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Carlos Sarah Elliott Henry I think the order may put the lesson inascendng order of difficulty.

Bq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Carlos No pref No pref No pref I would focus on Maria's first so the kidscould see exactly what they are doingwith objects rather than just numbers,then I Would teach them Carlos'sstrategy so they understand how to do iton paper.

Cq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Carlos No pref No pref I ranked my answers according to thestages in which I would like my studentsto learn multi-digit adding along withappreciating place values.

Dq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score CommentCarlos Maria Sarah Henry Elliott I think the simplest way was Carlos's

strategy, Maria's shows you hands-on,how it works, Sarah's would practiceestimation, Henry's strategy works andcould be useful, and i don't understandElliottt's strategy.

Eq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score CommentNo pref No pref No pref No pref Carlos I think it would be better to show how to

manipulate the number, how tounderstand the concept before showingthe algorithm. (Though I certainly wasnot taught that way and tend to rely toomuch on algorithms.)

B4–S3.3

Fq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Henry Sarah Carlos I think it is best to start out with visual aids that canhelp children to understand place value right off thebat. Elliott's, Henry's, and Sarah's thinking are allsomewhat similiar, and are various ways of solvingthe problem in one's head. Carlos's thinking is bestsaved until the end, when the children alreadyunderstand place value, and then the algorithm willbe of more importance and value to them.

Gq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Carlos No pref No pref A child has to begin somewhere and in order foranyone to learn you must begin small. Maria'sprocedure is a visual way to explain and lesscomplicated to learn for the first time.

Hq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Carlos Henry Sarah This seems like a logical order for me. I think it'simportant to understand the units before you moveinto the standard algorthim (Carlos' method). As apercursor to Carlos' method, it would be important tounderstand Elliottt's method, and before that Maria'smethod, which is essential. Sarah's method is morecomplex than Henry's, because it uses subtractionand number sense. Begnining with Henry's methodwould help encourage Sarah's to come about.

Iq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Sarah Henry Carlos I put Carlos last because I understand the way hehas chosen to do the problem. Yet with Maria, Iwould like to understand the blocks a little more. Ineed a refresher course on how to solve problemslike that. Then with the other students I would like tofigure out why they believe solving the problem thatway is the easiest for them, when it seems to makemore steps and more complications to arrive at theanswer.

B4–S3.3


Jq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Carlos Sarah I don’twant to

share anyothers.

I chose Maria first because when a childis learning math I feel it is always betterto show them a picture of themanipulatives that way they get anunderatnding of place value and not justhow to do the problem. Then I choseeliott because it goes along with Mariajust a little harder because you don’thav the picture, then I chose carlobecause he has a great idea, besidesafter they have place value down prettywell they will understand why you haveto carry.

Kq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Henry Sarah Carlos I think that first the student s must learnhow to do the p roblem visually so theyunderstand what they are doing. Thenthey can learn to do it with words andmanipulating numbers. Afterunderstanding that they can use Carlosmethod because they will understandwhat the columns mean and what theyare doing when they carry over anumber.

Lq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Sarah Carlos Henry I think it is important to start withmanipulative like with Maria’s approachand then go into written language.

B4–S3.3

Mq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score CommentCarlos Maria Sarah Elliott Henry Carlos is the most basic and should be taught

to all children, Maria’s is helpful for those whoneed visual support, and Sarah’s is helpfulbecause rounding is a fairly simple thoughtprocess for children. Elliottt’s and Henry’sways seem as if they would confuse childrenso I would probably leave these out of alesson and emphasize the other methods.

Nq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Carlos I don’t wantto shareany others.

Maria’s method seems a way to introduceaddition to get a base understanding and toget a visual. Then Carlos’ method makes itmuch faster and less work for the kids, oncethey have an understanding.

Oq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Carlos I don’t wantto share any

others.

The three that I chose were answered withgood logic. The ideas were clear and theirreasoning was also valid. They broke theproblem down into easier components, andwhen put back together, made sense.

Pq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Carlos Elliott Sarah Henry I would start out with the students that usedthe easy, plain way to see steps and the onesthat rounded the numbers. Then I would gointo a more advanced way of showing thenumbers by breaking them up in a wordsentence.

Qq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Henry Maria Sarah Carlos Elliott I rated them from the most confusing ways tothe least confusing way which they would beable to apply to other problems, in myopinion.

B4–S3.3


Zq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Henry Sarah Carlos I would want them to understand thevalues of the number before learningthe algorithim

Yq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Henry Sarah Elliott Carlos i would start out with the most beginnerfriendly and spend 2 weeks on that,then i would go to something a littleharder but still managable for a week,then i would go to something that helpsestablish place value for 2 weeks andthen i would conclude with the easiestway and the way that they will probablyend up using for the rest of their lives.

Xq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Henry Sarah Carlos i would introduce the "counting method"with manipulatives so that the childrenwould see the basic idea of addition andplace value. then i would move ontoworking problems out on paper andshow them an algorithim in which youare focusing on the values of theplaces. then i would intorduce henry'smethod of conceptualizing the placesand doing the math in their heads. afterthey catch on i would show them hownumbers relate to one another (withnumber sense) and create a classroomdiscussion with sarah's method. lastly iwould introduce the standard algorithimand hopefully they will not becomedependent on it, but rather use it insituations where an answer must befound quickly.

B4–S3.3

Wq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Sarah Henry Elliott Carlos I said maria was first because that wayyou can use the base blocks and showthat 10 singles equal a long and youcan change the singles in for a long,symbolizing carrying. I chose carols aslast because carrying the 10 is difficultto understnad expecially why and howto do it. Also adding in the 10 youcarried is confusoing because it makesthem wonder why and you can tehnjustify it with the base blocks.

Vq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Sarah Carlos No pref Maria is when they are still beginng tounderstand addition. Elliott I just loveand think it will be easier to understand.Sarah then can expalin what theyalready know. Carlos is kind of like ashort cut.

Uq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Elliott Carlos Sarah Henry Maria First Elliottt, because adding smallerand more rounded numbers is easier.Second, Carlos, because you have tohave the skill of basic addition the longway. Third, Sarah, Fourth Henry,because they have a similar type ofthinking. Maria fifth or possible earlier.

Tq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Carlos Sarah Elliott Maria Henry Carlos' method is obviously the easiestand quickest. henry got his problemwrong so i would not use it at all.Maria's method is too long

B4–S3.3

Sq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Carlos Sarah I don’t wantto share

any others.

I think the ranking shows a progressiveexplanation of the problem.

Rq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Carlos Elliott I don’t wantto share any

others.

First the children need to visualize theproblem before they on to doing it withjust numbers and Carlos’ way is moredirect than Elliottt’s or the otherchildren.

ZZq3.3_first q3.3_second q3.3_third q3.3_fourth q3.3_fifth q3.3_explain Score Comment

Maria Elliott Carlos I don’t wantto share any

others

Starting off with the blocks would beginwith a simple way. Then since we areworking on place value withmanipulatives, going into Elliottt’s wayto express value even more. Finallygiving them an algorithm, but knowingthey understand why they are using it.

B4–S3.3



A 1 Carlos second; difficulty matters, not concepts before standard algorithms

B 2 Carlos second; concepts (objects before numbers) before standard algorithms

C 3Carlos third; global comments indicate place value understanding before standardalgorithms

D 0 Carlos first; no need to look further in terms of this belief

E 3Carlos fifth; nice statement about concepts before standard algorithms, but no specifics areprovided.

F 4Carlos fifth; explains explicitly why she would teach Carlos' strategy last ("when the childrenalready understand place value").

G 2 Carlos third; order based on complexity, not on concepts before standard algorithms.

B4–S3.3


H 3Carlos third; explicitly describes the need to understand units (indicating place value) beforegetting to the standard algorithm.

I 2Carlos fifth; does not indicate a desire to teach concepts and then teach standardalgorithms.

J 3Carlos third; indicates a desire to begin with concepts prior to teaching the standardalgorithm

K 4Carlos fifth; indicates a desire to begin with concepts prior to teaching the standardalgorithm, and explicitly mentions that the students should understand underlying concepts(place value) before being taught the standard algorithm.

L 3Carlos fourth; very weakly indicates a desire to begin with concepts and then teachstandard algorithms. Because no explicit statement about understanding concepts prior toteaching standard algorithms is made, this response is a 3 (and a weak one at that).

M 0 Carlos first; no need to read the explanation.

N 2Carlos second; desire to teach concepts then standard algorithms indicated, particularlywith the comment “once they have an understanding.”

O 2Carlos third; although comments are made about the students’ reasoning, no indicationgiven of a desire to begin with concepts and then teach standard algorithms.

B4–S3.3


P 1Carlos second; seems to focus on easiest-to-most-difficult progression instead of onconcepts then standard algorithms.

Q 2Carlos fourth; does not indicate a desire to begin with concepts and then teach standardalgorithms (instead the response is focused on most confusing to least confusing).

Z 3 Carlos fifth; indicates concepts before standard algorithms but no details.

Y 3Carlos fifth; although some indication is given of concepts before standard algorithms(discussion of one of the strategies helping to establish place value), the response is notrobust.

X 4Carlos fifth; extensive detail about how each strategy relates to understanding the conceptof place value.

W 4Carlos fifth; conceptual progression between Maria and Carlos is discussed; the difficulty ofunderstanding Carlos's method is mentioned. Score is a weak 4 because only the one linkis mentioned between or among strategies to describe the conceptual progression.

V 2Carlos fourth; no explicit comment made about understanding place value before learningthe standard algorithm. No mathematical rationale given for any of the strategies.

U 1Carlos second; no explicit comment made about understanding concepts before standardalgorithms. The only comment is about Elliott's method’s being easier than Carlos's.

B4–S3.3


T 0 Carlos first; no need to read explanation.

S 2Carlos third; desire to begin with concepts and then teach standard algorithms is notindicated. The response is too vague to clarify to what “progressive explanation” refers.

R 2Carlos second; statement that children first need to visualize the problem (weakly) indicatesa desire to begin with concepts and then teach standard algorithms.

ZZ 3Carlos third; response indicates a desire to begin with concepts and then teach standardalgorithms. Note that although the respondent first discusses a “simple way,” she ends withan explicit statement about understanding concepts before learning standard algorithms.

Scoren % n %

0 87 55% 46 29%1 13 8% 11 7%2 32 20% 34 21%3 24 15% 43 27%4 3 2% 25 16%Total 159 159

Pre Post


B4–S9


Belief 4

If students learn mathematical concepts before they learn procedures, they are more likely to understand theprocedures when they learn them. If they learn the procedures first, they are less likely ever to learn the concepts.


This rubric focuses on the support that the respondent recommends the teacher provide to ensure that morechildren successfully solve fraction-division problems. In analyzing these responses, look for (a) the kind ofsupport the teacher might supply and (b) the timing of this support. Respondents who suggest that the teachersupply an explanation provide weak evidence of this belief. Although they show interest in conceptualunderstanding, they are insensitive to the timing of conceptual development. Respondents who suggest that theteacher develop the concepts by using visual aids, manipulatives, or story problems provide evidence of this beliefbecause they realize that more than an explanation will be required for children to understand this difficult concept.Those respondents who note that the teacher needs to supply the conceptual support before teaching theprocedure are considered to provide strong evidence of the belief. They realize the importance of providingconceptual support, and they note the importance of the timing of this support.

In coding this response, the coder should focus on responses to Item 9.6. Responses to other parts of the itemgive more information about the respondent’s thoughts on the importance and timing of conceptual learning.Coding this rubric is relatively fast, and interrater reliability is expected to be relatively high because little inferenceis required.





Yes No




B4–S9

Rubric Scores

0. Responses scored 0 include the view that the child needs only to practice more. Respondents may mention conceptualunderstanding, but do not describe it as something that is needed.

1. Responses scored 1 indicate that an explanation of why the algorithm works should be given to the child but do not include asequence for the instruction. Respondents state that the algorithm and concept should be taught concurrently. They offer noideas for how the concepts should be developed, beyond providing an explanation.

2. Responses scored 2 indicate that to understand the concept, children should have access to something, such as visual aids,in addition to an explanation of why the algorithm works. Respondents have an idea about how conceptual developmentmight take place, but they do not mention that it should come before the algorithm is taught.

3. Responses scored 3 indicate that before learning the algorithm, the child needs to have the conceptual foundation.Respondents suggest ideas for how this conceptual foundation should be laid (such as by using visual aids). They clearlystate that the conceptual work should come BEFORE the work with the algorithm.

B4–S9

Scoring Summary


0 • Practice more; no mention of teacher intervention of any kind OR• Practice more of the same kind with some support from the teacher.

1 • The teacher should explain why the algorithm works along with providing more practice.

2 • The teacher should provide a context, visual aids, or both.

3• The teacher should provide an explanation of the concept or experiences with the concept before

working with the algorithm so children can develop conceptual understanding. OR• Children can reason through this kind of problem and never need to learn the algorithm.

B4–S9

Examples

1q9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

The teacherdid a goodjob ofexplainingherself andthe child wsveryattentive.

I think sheunderstandsto divide youhave toreverse thesecondfraction, but Idon't thinksheundestandswhy she hasto.

Yes. She had agoodunderstanding and if shetried again ina few days itmay take hera little time,but I thinkshe'll get it.

She forgothow to do theproblem.

The child isvery typical,and without alot of practicethey will allhaveproblems.

Using anumber ofdifferentfractions insimiliarproblems.

0 The focus is on practice, with noindication that conceptual understandingshould be a part of learning mathematics.


it was goodthat sherememberedthat 4 equals4 over one.

I think thechildunderstandsthemathematicalprocess ofhow it's donebut she maynot be able tounderstandwhy you flipthe fraction tomultiply orwhy 4 isequal to 4over 1.

No i don't thinkshe would beable to do itin three daysunless shewent homeand practicedit more.

she didn'trememberhow to solveit.

A few goodstudentsmight havevery goodmemories fornumbers.

providehomework orwork on it inclass everyday until thechild is ableto rememberon their own.Anexplanationof why it isdone the wayit is wouldhelp too.

1 Need for conceptual understanding asbasis for the algorithm to be learned isacknowledged; however the way todevelop such understanding does not gobeyond an “explanation of why.”

B4–S9


The little girlknows how tosolve theproblem aftershe practicedthe step sheneeded to doin order tosolve theproblem.


Yes She would beable to solveit on her ownbecause shealreadypracticed theproblemenough times

The childcould notfollowthorughbecause shedid not havean exampleto follow .

There is afew childrenwho couldnotremmemberhow to solvethis becausethey mightnot have agood memory


2 Need for conceptual understanding isnoted; respondent suggests involvingsomething (in this case, a picture)beyond an explanation of why but doesnot specify that conceptualunderstanding should come first.


The studentdidrememberthe correctalgorithm butI don't know ifsheunderstoodwhat she wasdoing or whyshe flippedand thenmultiplied.Where wastheunderstanding.

I think sheonlyunderstandshow thealgorithm issupposed towork. I thinkshe onlymemorizedthe stepswithout anymeaningbehind them,

No I havelearned thatsimplymemorizinghow to dosomething isnot a goodway torememberhow to dosomething.Youmustunderstand howsomethingworks andwhy it worksin order touse it later.

She had onlymemorizedthe algorithmof dividingfractions for ashort time.She had nounderstanding of what todo with thenumbersbecause thealgorithmgave her nomeaning asto what shewas actuallydoing.

I only thinkthat thosefew childrenwith reallygoodmemorieswould knowhow to do itthree dayslater.

The teacherneeds to givethe students

meaningbehind thealgorithm.She couldshow them

with picturesor blocks first

and thenshow them

how thealgorithm ties

in with it.

3 Respondent emphasizes conceptualunderstanding and the importance ofconceptual development prior toexposure to an algorithm.

B4–S9


Aq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

Impressive,the childseemed tounderstandwhat wasbeingexplained toher. She didnot guess butlistenedattentively totheinstructions.

I think thatsheunderstandthe concept.

Yes she seemedto have agood grasponcesomethingwasexplained toher.

The childcould notrememberwhat hadbeenexplainedbefore.

'Fractions arethe hardestand mostconfusing tolearn

Continuereinforcing theconcept.

Bq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

Sherememberedthe steps insolving theequation, andrememberedwhat theteacher hadtold her. Shewas able tolearn themand applythem to otherproblems.

I dont thinkshe quiteunderstandswhy you flipit, but that itssomethingyou do tosolve theproblem.Sheunderstandsthat you flipthe fractionand thenmultiply itstraightacross.

Yes I haveconfidencethat this childcould solvesimilatproblemsbecause sheknows thesteps andmethods ofhow to goabout solvingsuch aproblem

She forgotthe steps ofhow to solvethe problem.

On furtherthought, if theprocess isntpracticedoften, it canbe forgotten,especiallysince its justlearned.

Explain whythe problemworks the wayit does.Possibilycreate asenario toexplain whatis going on..orto use morevisual aids.Also, assignhomework onthe subject sothe studentscan continuelearning theprocess andmemorize it athome, andthen teach itmore in class.

B4–S9

Cq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

i think thatthe child inthe clip withpractice canlearn themethod ofdividing andmultiplyingfractions.

nothing qutieyet, maybeplacing a oneunder awholenumber .

Yes if shepracticed themethod shelearned fromher teacher.

she struggledonrememberingwhat to dowith thefraction as faras what to dowith the oneand thethree.

i would saythat less than50 percentcould solvethis a fewdays later. itwould takepracticebecause it isdealing withfractionsrather thanjust wholenumbers.

try to explainmorethoroughly onwhat to doand why thedivision signturns into amultiplicationsign whenputting a oneunder thewhole numberand allow thechildren topractice more.

Dq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

The child wasable to pickup thisconcept veryeasily and itwasexplainedvery wellalso.

It ismultiplyingwith a flip

Yes. The childknew how tosolve theproblem andshe couldexplain it.They say thatif you canexplain it, youreallyunderstand it.

The child wasnot practicingher divisionof fractionsand forgotthe process.If you do notpractice, youwill notremember.

I think thathalf of thechildrenwould beable to solveit a few dayslater and thenthere will bethe ones whostruggled.

Assignhomeworkrelating to thisso they canpractice.

B4–S9

Eq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

I was verysurprised tosee that afterseeing threeexamplesand onlydoing oneproblem onher own, sherememberedall the stepsto solving theproblem.

I think thechild onlyunderstandswhat theteacher hasshown her todo. Sheprobablydoesn'tunderstandwhy.

Yes. I think thechild wouldbe able tosolve asimilarproblem in afew days, aslong as shehad sometype ofhomeworkinvolvingthese typesof problemsor some typeof review.

The childcould notrememberthe steps tofiguring outthe answer tothe problem.She wasobviouslyonly shownthe few daysbefore withnoexplanationof why andnot given anymore practiceat this type ofproblem.

I would sayabout 5because notmanychildren willrememberthose stepsto solving theproblemwithoutunderstanding why theyhad to dothose certainsteps.

If the teacherasked thechild morequestions andhad given thechildexplanation asto why thosesteps tookplace, I thinkmore childrenwouldremembermore.

Fq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

confusing tochild, boring,meaningless

nothing No she was verygood atmemorizingthe algorithmin the shortperiod oftime, but itwill not staywith her.

painful towatch. Childhad nounderstanding of what wasbeing asked.Child took along time towrite the 6because sheknew fromthe beginningthat shedidn't knowhow to do theproblem.

nounderstanding of what thedivision reallymeant

I don't knowhow you showdivision withmanipulatives(yet) but youwould have tostart that way.The childrenwould have tohave manyexperiencesto understandthe conceptbefore evenshowing themthe algorithm.

B4–S9


Gq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

the child didecactly whatshe was told,no more orno less. sheseemedpassive anddidnt askmanywuestions.

i dont know ifshe knowswhy she flipsthe fraction orwhy shemultiplies. idont thinkshe reallyunderstandwhat she isactually doingwith thefractions.She wasgoing throughthe steps likeshe watchedthe teacher. ithink that shesolved theproblembecause ofwatching theteacher.

No. she willprobablyforget themethod andwill need theteacher toshow herhow to do itagain.

the childforgot themethod shewas shown afew daysago. thisshows thatshe didntunderstandtherelationshipof the 2fractions andwhat she wastrying tosolve.

about 20% ofthe childrenwould beable to solvethis problemlater becausethey weretaught tooquickly onhow toperform theproblem, nothow tounderstandthe problem.There wouldbe a few thatwouldrememberhow to do theproblembecasue theygot a betterunderstanding or theyhave a bettermemory.

the teacherneeds to drawvisuals ormake up astory so thechildren canrelate to theproblem. Ifshe explains itin a way thatthey arefamiliar with,the numbersand thesymbols wontbe so foreignto them. theycouldremember thestory to guidethem insolvingproblemssimilar to this.

B4–S9

Hq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

the processis explainedrather thanthe concept

nothing No, she might ifsheremembersthe processbut shedosentunderstandthe process

concept isnotunderstood

a greatmajority willbe able to dothis if giventhe steps butnone willunderstandwhat thedivision offractions areand why thesteps whythey work

explain theconcept firstthen show thesteps

Iq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

nothingmuch,

that you justtake therecipical ofthe fractionand themultiply them

Yes b/c theteacher did agreat job atdiscribinghow to do it

she didntrememberwhat to do

' think she isquite typical,but I think60% ofchildrenwouldrememberhow to do it

maybe do theproblem moreor come upwith a littlesong ormelody tohelp out withrememberingthesteps

B4–S9

Jq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

there shouldhave beenway morebuilding up tothis lesson.the girl knowshow to gothrough themotions but idon't thinkshe reallyunderstandswhat she isdoing. shedoesn't evenhave anyidea of whatdividingfractionsreally is. orwhat she isdoing

absolutelynothing, butshe can solvevery basicones. that isgreat that sheknows analgortithm butin a weekfrom now iwonder if shewillremember it.

No if you don'tunderstandwhy are whatyou are reallydoing, itmakes itdifficult torememberbecause younever reallyknew whatyou weredoing in thefirst place.

interesting. ibet if therehad beenmore of awhy lessonand shereallyunderstood ita couple ofdays ago,she wouldhaveremembered.instead shenever reallyunderstood itand that isapparent inher nothaving anyidea on howto solve it afew dayslater.

i am not surehow manywouldremeber, buti would thinkit would be avery smallnumverwouldremember

have more ofa conceptuallesson beforethe algorithm.i think it is okto showstudentsshortcuts afteryou know thatthey reallyunderstandwhat they aredoing. if therehad maybebeendrawings,patternblocks, orsome type ofmanipulativeto help the

B4–S9

Kq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

the fact thatthe teachershowed herexplanationon paper wasvery useful

im notcompletlysure is sheunderstandsthe processor is simpleimitating it.

Yes with som help This showsthat the childsimplyimitated whatshe saw theteacher doingbut did notunderstandthe processshe wasusing to getthe answer.

Unless theyunderstandthe conceptsused andpracticesolving moreproblemsthey wontknow what todo.

explain to thechild exactlywhat goingon, dont justmake herimitate youractions.

Lq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

The thing thatstood outmost washowconfusing thealgorithmseems to be.If I were thelittle girl Iwould bethinking whydo we dothis.

I do not thinkthat the childunderstandsmuch aboutfractions.

No She will haveprobablyforgotten thesteps used tocomplete theproblem.

She could notremeber howto do theproblembecause themethod shewas taughtmade nosense to her.

I do not thinkthat manychildrenwouldrememberhow to solvethis problembecause itwould notmake anysense tothem.

Explain whywe solve theproblem usingthis algorithmand alsoexplain to thechild differentways of doingthe sameproblem. Forexample usingmanipulativesor storyproblems.

Mq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

it was a goodteachingapproach

that in orderto devidefractions youmust flip thesecondfraction overand thenchange it totimes.

Yes probably ifshe hadthose notes

it wasinteresting tosee that shecould notremember

I think she ispretty typical

give her littlelessons alongthe way

B4–S9


Zq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

The childmemorizedthe formatused by theteacher. Sherepeated theexact samemovementsandattemppts asthe teacher.

There's noway to tellunless yougive her twocompletelydifferentfractions todivide.

Yes A child'smind is like asponge inthat it canabsorb mostknowledegewith a clearview ofunderstanding.

She simplyforgot theprocedures ofsolving theproblembecause thenumbersslightlychanged.

With morepractice andfurtherexplanationthe methodwould sinkinto thechild's mindabling themtobe successfulin the future.

Have patienceandcontinuallyexplain andprovidesuggetions tothe studentthat may helpthemremember orbetterunderstandthe procedureat hand.

Yq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

the child onlymemorizedthe solvency,but was notactuallythinking onwhat she wasdoing

not much,only theprocedure

No as I said, shewas justfollowing theprocedure,but that doesnot mean shewasunderstanding it

The little girldid notrememberwhat to do.she probableknew shehad tochange thedivision signfor other signand that iswhay she putthe plusinstead of thetimes.

if they did nothave anypractice afterthe firstsession, thennone wouldbe able toremember

As I haveseen in manyvideos fromclass 29.6, Ithink theteachershould firstteach themwith drawingsormanipulatives, and then thealogarithm

B4–S9

Xq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

The child didwell onworking theproblemalone. Butone thing thatstood out tome was howwell did sereallyunderstandthewhole'flipping overandmultiplying'thing.

Sheunderstandshow to gothrough themethod andget the rightanswer.

No Withoutpractice no.She has onlyworked a fewproblems andshe is goingby metho, notbyunderstanding.

The child didnotrememberthe processafter a fewdayswhichshows justwhat Ithought, shehas nounderstanding of theprocess justa shortmemory howto copy theteacer'sexamples.

This child isvery typical,with nounderstanding behind aprocess veryfew would beable toremember afew dayslater

Work on theunderstanding of fractions,what divisionof fractions isall about.Also work onthe reciprocalof frations,giveexampleswhy therecipricalmultipliedworks thesame as theoriginaldivided.

Wq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

yes - whenasked if shewanted tolearn shedidn't sayyes... otherthan that hersniffling in themic... :) yesshe figuredout how tocomplete theproblem - ididn't expectthat

how to solvethe problem -but not whyYes

Yes if she has agoodmemory.....

she wasconfused and icould tell shefelt dumb withthe camera andthe teacherwaiting... sheknewsomething wassupposed toflip... and sheknew the signwas supposedto change.. butshe couldn'tfigure out inwhat order orwhere

it dependson theirmemory.. ifa child has agoodmemory theycould justcopy cat theorder withdifferentnumbers...howevermost kidsmemorygoes topokemonand barbie

more teachinandpractice...unless shewants toexplain why- but i doubtthe childwould fullyunderstand

B4–S9

Vq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

i liked thatthe teacherfirst showedher how to dothe problem,however itthought theteachershould haveexplainedwhy we docertain steps.this helps thechild not inthe waytheat shewon't dootherproblemswithmemorizingsteps

i think sheknows thesteps you gothrough insolving adivisionproblem offractions

Yes bymemorization

i think that ifshe was toldthe reasonsbehind whyyou do thesteps shewould haveunderstoodbetter.howeveri likethat theteacherasked whatare youthinking andleft it at that ,didn't feel theneed for thestudent to getthe answer

I think halfwould beable to solveit. not waitthree days,maybe dothreeproblems aday withregularsession.

not wait threedays, maybedo threeproblems aday withregularsession

Uq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

The little girlknows how tosolve theproblem aftershe practicedthe step sheneeded to doin order tosolve theproblem.


Yes She would beable to solveit on her ownbecause shealreadypracticed theproblemenough times

The childcould notfollowthorughbecause shedid not havean exampleto follow .

There is afew childrenwho couldnotremmemberhow to solvethis becausethey mightnot have agood memory


B4–S9

Tq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score CommentThe teacherworked withthe child andfirst showedher how to dothe problemand thenallowed herto do it on herown wihtsome help.

Sheunderstandthat you haveto flip thefraction andmultiply inorder to solvethe problem.

Yes Since she didit a few timesI think sheshould beable to solvethe problemor at least geta start on theproblem.She mighthoweverneed themodel toguide herthrough thesteps again.

The child hasforgotten howto do theproblembecausethere is nomodel andthis is a newconcept forher.

This is prettytypically for achild who isjust learninghow to do anew problem.it takes timeand practisefor them tobe able tounderstandwhat theredoing and doproblems ontheir own.

I think theteachershould helpthe child andtry to refreashher memory insomeway.This child leftwithout reallyknowing howto solve theproblem.

Sq9.1 q9.2 q9.3_choice q9.3 q9.4 q9.5 q9.6 Score Comment

The nevershowed thegirl withdrawingswhat theywere doing.The couldhave drawn 4squares, thendevided theminto thirdsand countedup thenumber ofthirds therewere.

I think tha thechildunderstandshow todevide themusingnumbers, butshe has noidea why sheis doing it.She has nomeans ofapplying thedata toeveryday live.

Yes The videosaid that sherehearsed afew moreproblems.Hopefully shewill havememorizedthe process.

She hadforgotten howto devide withfractions. Iaslo don'trememberseeing a timeframe, wasthis the nextday? A weeklater? Amonth?

Her matchskills are notout of thenorm.

I think thatshowingillustrations ofwhat is beingdone orprovidingmanipulativeswould havehelped.

B4–S9



A 0 The emphasis in this response in on practicing the algorithm.

B 2Respondent mentions that explaining why the algorithm works and suggests that visualaids be employed to illustrate the ideas.

C 1Respondent states the importance of practice; score is 1 because of the comment that theteacher should explain what to do.

D 0 Mentions only practice.

E 1Respondent suggests that explanation of steps of the procedure will promoteunderstanding.

F 3Respondent notes that this concept should be developed through use of manipulativesBEFORE instruction on the algorithm. The time element is critical in the score of 3.

G 2Respondent advocates helping children understand the concepts by using visuals andstories rather than number symbols. She does not mention that this work should comebefore the symbol work, so this response is scored 2.

B4–S9


H 3How the concept will be explained is not specified, but the respondent’s comment thatexplanation should precede showing the steps is interpreted as evidence of this belief.

I 0 Focus is on practice.

J 3That the conceptual lesson should come before the procedural lesson is mentioned inresponses 9.1 and 9.6.

K 1Understanding the concepts is important, but one can come to understand through acomprehensive explanation.

L 2The teacher should use manipulatives and story problems at the same time that sheexplains the reasoning behind the algorithm.

M 0The meaning of “little lessons” is unclear but was interpreted as a review of the stepsinstead of as an explanation of the concept.

Z 1The notions of “further explanation” and “better understand the algorithm” are vague butwere interpreted as indicators of a desire to explain why the algorithm works.

Y 3This respondent recommends developing the concepts through multiple representationsbefore teaching the algorithm.

B4–S9


X 1The meaning of “work on the understanding of fractions, what division of fractions is allabout” is unclear without further details; it was interpreted as a suggestion for verbalexplanation of why the algorithm works, in conjunction with the algorithm.

W 0Although the participant expresses an interest in explaining why, he or she is clearlydoubtful about the likelihood that such explanation will help.

V 1Explain why with more practice (note that this explanation is in response 9.1, not response9.6).

U 2Although overall the response seems unimpressive, the “maybe with pictures” warrants a2.

T 0 Practice is advocated with no mention of concepts.

S 2 Because of the mention of illustrations and manipulatives, the response is scored 2.

Scoren % n %

0 96 60% 55 35%1 42 26% 47 30%2 18 11% 46 29%3 0 0% 11 7%Total 156 159

Pre Post


B5–S2


Belief 5

Children can solve problems in novel ways before being taught how to solve such problems. Children inprimary grades generally understand more mathematics and have more flexible solution strategies than adultsexpect.


Each of the two parts of this segment is coded separately; the two part scores are then considered to determineone overall score (see the Scoring Summary).

Responses for this segment were often found to be ambiguous and difficult to code. Scores depend uponwhether respondents are sensitive to children’s thinking and view children’s approaches as legitimate. Somerespondents, however, provide little information on this subject.

2. Read the following word problem:

Leticia has 8 Pokemon cards. She gets some more for her birthday. Now she has 13 Pokemon cards. How many Pokemon cards did Leticia get for her birthday?

2.1 Do you think that a typical first grader could solve this problem? NOTE. The problem could be read to the child.

Yes

No

You answered that a typical first grader could solve the following problem:


2.2 If a friend of yours disagreed with you, what would you say to support your position?

Here is another word problem. Again, read it and then determine whether a typical first grader could solve it.

Miguel has 3 packs of gum. There are 5 sticks of gum in each pack. How many sticks of gum does Miguel have?


Yes

No

B5–S2

Rubric Scores

0. Responses scored 0 indicate that children will not devise legitimate approaches to the Join Change Unknown andmultiplication problems and thus provide no evidence for the belief that children can devise their own solutionsbefore being shown how to solve problems.

1. Most responses scored 1 indicate that children’s approaches to only one of the two problems will be legitimate.Responses scored 1 do not indicate sensitivity to children’s thinking, although some indicate that children cansolve both problems once they have been taught. These responses are interpreted as providing minimal evidencethat children might be able to solve problems for themselves, but they indicate primarily that children must beshown how to solve the problems.

2. Responses scored 2 indicate that children can generate solution strategies for one of the problems and must betaught the other. They indicate that children can solve some problems independently.

3. Responses scored 3 indicate confidence that children can solve both problems when they have access tomanipulatives. They indicate the respondent’s faith in children’s abilities to devise their own solutions.

B5–S2

Scoring Summary

Use the following tables to score Items 2.2 and 2.4 separately; then calculate the overall score.

Score Rubric details for Response 2.2

0 • Simple subtraction problem—children have already learned how to do it.

1 • Cannot solve (or comprehend) problem OR Can do it if they know that it is subtraction.

2 • Can do it after being taught how to model this problem

3 A. Typical children can do it if they have manipulatives OR I have seen it. B. This is a hard problem for children when they invoke their informal knowledge, because it is a

missing-addend problem.

Score Rubric details for Response 2.4

0 • Can solve but child’s approach is not legitimate

1 • Cannot solve

2 • Can do it after they have been taught

3 • Typical children can do it if they have manipulatives. OR The child can add 5 + 5 + 5.

Score permutations forOverall score (2.2, 2.4)

0 (0,0) (1,1) (2,1) (1,0) (0,1) (2,0) (0,2)

1 (3,1) (3,0) (2,2) (1,3) (1,2) (0,3)

2 (3,2) (2,3)

3 (3,3)

B5–S2

Examples

1q2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I would say that 1st

graders are alreadytaught how to add andsubtract. This is aneasy subtractionproblem.

No This kind of probleminvolves the concept ofmultiplication.Multiplication is notsomething that isintroduced to a child infirst grade. Even ifsomeone might say thatthe child may be able toadd 5 three times, it isthe concept ofmultiplication that allowsone to solve thisproblem, and a firstgrader is not there yet.

0 0 0 The 2.2 response fails toshow sensitivity to children’sinterpretations of this as aJoin Change Unknownproblem. In 2.4 therespondent suggests thatfirst graders may be able tosolve this problem but thatthey are not reallymultiplying.

2q2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo I believe that not all

children can think ofthe answer on the topof their head. Somechildren might getdiscouraged by thenumber 13.Sometimes childrencannot count thathigh.

No I think that it would bedifficult for the child tograsp the idea that theyare multiplying. Theymight just see the 3 and5 in the word problemand add them up.

1 1 0 In both responses therespondent expresses doubtabout first graders’ abilitiesto solve the problem.

B5–S2

3q2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I would tell her that

you could set up theproblem like this: 8 +_ = 13. Then I wouldtell her that all youhave to do to showthe first grader how tosolve it is to showthem that this is alsoa subtraction problem:13 – 8 =

Yes I think that if a child istaught to do a problemlike this one, he will beable to do it, no matterwhat his age is.

2 2 1 Both responses indicatethat children will be able tosolve these problems oncethey have been taught.

4q2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I would argue that a

first grader would havethe knowledge in addingand subtracting enoughto be able to count fromeight to thirteen onusing their fingers theywould be able to seethat she got five newcards.

Yes they can draw out theproblem. they woulddraw 3 groups with fiveobjects in each group.then count all theobjects.

3A 3 3 The respondent expressesconfidence in children’sabilities to solve bothproblems. The responsesshow a sensitivity tochildren’s intuitiveapproaches to theseproblems.

5q2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score Comment

No It is a missing addendproblem and that canbe harder for youngchildren to solve than astraight addition orsubtraction problem.

Yes By using manipulativesthe child will be able tosolve the problembecause it easy forthem to see what theyare doing.

3B 3 3 In 2.2, the respondentshows sensitivity to thedifficulty of this kind ofproblem. In 2.4 she notesthat multiplication is easyto model.

B5–S2


Aq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo In first grade children are

dealing with adding numbersand place value so somemissing number problemsmay be too tough for them.

Yes well, this is something thatthey can draw out so theycan visibly see the fivesticks in each pack. it maytake some time before theanswer is reached but theycan handle it.

Bq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo It is basic algebra, and

unless they are taught thetheory behind algebra, theywill be confused. Maybe afirst grader could figure it out,but it would take a lot ofthinking and a lot of hands-on.

No This is multiplication andgenerally students do notlearn this until third grade.I also think that it is a verydifficult problem to a firstgrader and that they ouldnot even be able tounderstand the prblem, letalone, solve it.

Cq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I would say that a first grader

could figure it out bysubtracting 8 from 13.

Yes I would show the firstgrader with objects andhelping him/her count

Dq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes i would say that if she had

cards Maria would beable tocount them to determine theamount

No This is at a higher level ,as this would involvemultiplication, most at thisage are unaware of this

B5–S2

Eq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I wouldn't argue with her

much because at thismoment I can't rememberwhat level of math I was atin the first grade. I knowthat all children learn atdifferent rates but I can'tremember the grade where Iwas first taught additionand subtraction.

No I would say that I am prettyconfident that unless thischild has had teachingoutside of school, he has notlearned how to multiply ordivide at this stage in hisschool career. I believe thatcomes closer to third andfourth graders.

Fq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo that it is not obvious to first

graders how to solve thisproblem. It would takesome help for them tofigure it out.

Yes because although they don'tteach 1st graders multiplicationthey would be able to count 5sticks of gum 3 times.

Gq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I had this exact

conversation with a firstgrader at a school I workedat.

Yes First graders know how to userepeated addition forproblems older kids could usemultiplication for.

B5–S2


Hq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score Comment

Yes I think first graders if giventime and can hear and seethe problem in most it willregister, but for those whostill struggle why not bring outthe amount of cards in theproblem and let them visually(hands on) figure it out; ifgiven the right tools

Yes this problem would be harderthan the previous due to factsome children struggle withmultiplication, but again Ibelieve if those who struggleare given some visual aidthey to can determine thecorrect answer

Iq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes its easy to see that she

started with 8 and now she has13 so the child would know toadd to 8 until they reached 13,or just subtract 8 from 13

No most first graders don'tknow how to domultiplication problems

Jq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo I do not know much about

how much a first graderknows. I would tell themabout my cousin who is inkindergarten and that he doesnot know how to add yet

No This is multiplication and thatis not taught until the 4th or5th grades

B5–S2

Kq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I would say that if you read

the problem to the child andhad visual aids to show whatthe problem asking then thechild might be able to seewhat was going on andanswer the problem with thecards that you set out.

Yes I think that with the teachershelp the student could solvethis problem. But a first gradercould not do this on his/herown. The teacher or helperwould have to guide themthrough the step for solvingthe problem. They would readthe problem, get out 3 packsof something and then put 5things in each pack and havethe child count them and see ifthey could figure out theanswer.

Lq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes as long as the child can

count, and there is norestriction on the tools thatthe child can use it would besimple for a firt grader to beshown eight of somethingthen a certain numer to getthirteen.. the child wouldneed to figure out how manywere added

Yes if the first grader uderstoodaddiion then it is simplycounting and adding.. i amnot sure at what levelmultiplication is introduced

B5–S2

Mq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes Well I would explain that most

children learn better withvisuals. As long as thechildren know how to count Ican show them the cards andthey can just count how manyI have added or subtracted. Ithink by them actually seeingthe cards it will help them tobetter understand.

No This problem requiresmultiplication, and althoughI'm sure some first graderscould get it I feel thatmultiplication is a conceptthat should be delt with at anolder age. First graders arejust learning addition andsubtraction.

Nq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo i would say that for a typical

first grader, it would be hardfor the child to hold on to twodifferent values at the sametime, meaning the start andthe result, not knowing thechange that occcured.

Yes i think that a typical firstgrader would be able to solvethis because they couldvisualize what they actuallyhave, meaning the 3 packs ofgum, that have 5 pieces each

B5–S2


Zq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes I would say that children have a

good sense of the things theyhave, and how many of themthere are. Children havememories that work quite well,especially when it comes totoys, or their prizepossessions. They know howmany they have, and many ofthem know how many toys theydon't have in a set of possibletoys to be bought from anymanufacturer. Therefore theywill know at a very early agehow many they received, andoften who a gift came from.

Yes I would say that every firstgrader knows how to count to15, and has had packages ofgum with 5 sticks in it manytimes by the first grade. Itmight take some children longerthan others, and perhaps somechildren wouldn't get it, but Ithink a teacher can spendenough time to get a child toimagine and picture thescenario in their minds, andcome up with acceptableanswers to gum quantities aslong as the numbers are not toolarge.

Yq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo a typical first grader is about

six years old. He or she may ofcourse be able to count but itmight be difficult for he or sheto solve this problem. If theproblem said maria has 8pokemon cards and received 5more for her birthday, how manydoes she have now? i believethis question would be easierfor a first grader.

Yes because i believe that a typicalfirst grader can count, threepacks of gum with five sticks ineach one without any missing, atypical first grader could answer.simply because five is an easynumber to count by. they canlay out each pack, know thatthere is five in each one andcount from there.

B5–S2

Xq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes i believe that children have the

ability to take a word problemand convert it into a regularaddition or subtraction problem,especially when it involvesobjects that the children arefamilar with, like pokemon cards

No i'm pretty sure that a first gradercan not solve a word probleminvolving multiplication, i believeit may be too complex of aproblem for that age group.

Wq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo First graders are at the

beginning of learning theirreading skills. They would needsomeones else to read thisproblem to them. I think it isharder to comprehend when youcan not read it to yourself a fewtimes. The first grader wouldhave trouble picking out all theclues to solve this problem byonly using their listening skills.

No Again, a first grader is at thebeginning level of reading andwould have troublecomprehending the problem.This problem also involves ahigher level of thinking that theyhave not learned yet

Vq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentYes you could bring cards into the

class and demonstrate it. youcould also set up the problemas 8 plus what equals 13. thisproblem actually might be a littletought for a first grader.

No i don't remember doingmultiplication in first grade andthat is what this probleminvolves 5x3= 15. i think firstgraders can only handle addingand subtracting

B5–S2

Uq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo A typical 1st grader would have

trouble reading the problem,because they are still learningto read. If the problem wasread to them, it might make iteasier. I think it depends onthe child. It seems thatchildren have trouble with wordproblems especially

Yes If read to them, the childrencould figure out the problem, butit does seem a little advanced.They don't know their timestables yet, but they could uselines to count and firgure it out.

Tq2.1 q2.2 q2.3 q2.4 2.2 Score 2.4 Score Overall Score CommentNo I think the first grader would

have to see pictures tounderstand because only a fewreally can hear and understandthat 13 is after the 5 hat weregiven. A child sometimesthinks that 13 were given.

Yes I think in this circumstance thechild could get messed up butthey can relate with a pack ofgum and he sticks inside. andthen add them up, but certainlynot multiply.

B5–S2


Exercise2.2

Score2.4

ScoreOverallScore

Comment on 2.2 Comment on 2.4

A 3B 3 3

This kind of problem can be difficultfor children.

Children will be able to easily solvethis problem.

B 1 1 0

This is an algebra problem. This problem will be too difficult forchildren

C 0 2 0

Child will use subtraction Will teach the child

D 3A 1 1

Case made for how children will beable to solve the problem by acting itout

First graders will be unable to do thisproblem.

E 0 1 0

Problem is simple subtraction. Children will be unable to solve themultiplication problem.

F 2 3 2

Children will be able to do this withhelp.

Children will be able to do thisproblem by counting.

B5–S2

Exercise2.2

Score2.4

ScoreOverallScore


G 3A 3 3

Has seen children do this kind ofproblem.

Children can model this problem withmanipulatives.

H 3A 3 3

Children can do these problems ifthey have visual aids to help them.

Children can do these problems ifthey have visual aids to help them.

I 3A 1 1

Difficult to code because bothaddition and subtraction arediscussed. The spirit of thisresponse is that children will solvethis by adding on, in keeping with thespirit of the 3 score.

Clearly scores 1.

J 1 1 0

Children will not be able to solvethese problems.

Children will not be able to solvethese problems.

K 2 2 1

Once they have been taught, childrenwill be able to solve these problems.

Once they have been taught, childrenwill be able to solve these problems.

L 2 3 2

Coders disagree whether to scorethis response 2 or 3

Can be done by adding

B5–S2

Exercise2.2

Score2.4

ScoreOverallScore


M 2 0 0

“Can show them” indicates that theteacher will need to show childrenhow to model this problem.

Although some first graders might beable to solve this problem, we inferthat the respondent does not valuetheir approaches.

N 3B 3 3

Difficulty of the Join ChangeUnknown problem is noted.

Children can solve the multiplicationproblem with manipulatives.

Z 3A 2 2

Good rationale, although nospecifics, given for the child’s beingable to solve the problem.

Some children could solve theproblem; all could solve it once theyhad been taught.

Y 3B 3 3

This Join Change Unknown might bedifficult for children.

Multiplication will be easy to solvebecause the child can easily act itout.

X 3A 1 1

Could be scored more than one way.Statement that child can easilyconvert this into a “regular” problemdoes not show sensitivity tochildren’s thinking. That the contextwill support the child’s thinkingshows sensitivity. Because of faith inthe child’s ability to do this problem,credit given for sensitivity.

Doubt that children can solve themultiplication problem

B5–S2

Exercise2.2

Score2.4

ScoreOverallScore


W 1 1 0

Children could not solve. Distractionof reading issue, but children cancomprehend story problems whenthey hear them. Seems to show littlefaith in children’s intellectualcapabilities.

Children could not solve; readingissue; seems to show little faith inchild’s intellectual capabilities.

V 2 1 0

Children have to be taught how to dothis problem.

Multiplication will be beyond childrenof this age.

U 1 3 1

Child will struggle with this problembecause it is a word problem thatshe will have trouble comprehending.

More optimism expressed about thechild’s chance for success for themultiplication problem.

T 1 3 1

Children will have difficultycomprehending this problem.Unclear whether reference is to thisparticular problem type. The generalskepticism expressed aboutchildren’s abilities contributes to thelow score.

Children could “mess up” on themultiplication but could do it bymodeling the problem.

Scoren % n %

0 74 47% 36 23%1 62 39% 60 38%2 5 3% 6 4%3 18 11% 57 36%Total 159 159

Pre Post


B5–S5


Belief 5

Children can solve problems in novel ways before being taught how to solve such problems. Children in primarygrades generally understand more mathematics and have more flexible solution strategies than adults expect.


Because this segment, unlike most others, does not pertain to a specific domain in mathematics, the responses tendto be more varied than for most other items. We identified eight response types that are represented in the rubric. Otherresponses fit none of our categories.

The focus of this rubric is faith in children’s thinking. Coders are expected to attend to whether respondents expressconfidence that children will be able to devise legitimate solution strategies when confronted with problems of typesnew to them. Responses that indicate doubt receive lower scores.

5. What were your reactions when you were asked to solve a new kind of problem without the teacher's showing you how to solve it?

5.1 When you are a teacher, will you ever ask your students to solve a new kind of problem without first showing them how to solve it?

Yes

No

5.1 You answered that you would ask your students to solve a new kind of problem without first showing them how to solve it. Please elaborate on your reasons:

5.2 How often will you ask your students to do this?

B5–S5

Rubric Scores

0. Responses scored 0 include an indication that children must be taught how to solve problems before they can solve themon their own. They indicate that asking children to solve problems without first showing them how to solve them isdamaging to children and that solutions of children who have not been shown how to solve a problem will be incorrect.

1. Responses scored 1 include doubts about whether children will be able to solve problems without first being shown howto solve them. The respondents express willingness to present children with problems but also express doubts, makingsuch comments as “to see if they can solve it.” The doubt in these responses takes several forms: (a) The children maybe unable to devise solutions; (b) providing problems will stimulate children’s thinking, but the children are not expected todevise solutions; (c) only a very few children will devise solutions; (d) children will devise various solution strategies, butthey will engage in this practice very rarely. Generally, the solutions will not be evaluated, or children will not be asked toshare their answers.

2. Responses scored 2 indicate that children can devise solution strategies when presented new problems. The responseslack expressions of doubt and are affirmative in nature. They often include a rationale for asking children to solve problemswithout showing them how to do so.

3. Responses scored 3 include positive statements that children can generate their own solution strategies to novelproblems. Respondents suggest that they plan to depend on using novel problems as a focus of instruction and expressfaith in children’s abilities to devise solutions for themselves.

B5–S5

Scoring Summary


0 • Children will not know what to do with the problem OR children’s initial work will be incorrect ORboth.

1

Children will be asked to solve problems without being shown how to solve them;• (a) but doubt in children’s ability to solve problems is expressed.

• (b) doing so will stimulate thinking (but no mention that solution strategies will emerge).

• (c) but only a very few children will be able to solve the problems.

• (d) children will generate their own approaches but will do so rarely.

2• Children can solve problems for themselves and will definitely devise solution strategies—but

not so reliably as to be the focus of instruction (i.e., one cannot depend on children’s thinking asa basis for instruction).

3 • Building on children’s thinking will be an integral part of instruction (i.e., one can depend onchildren’s thinking as a basis for instruction).

B5–S5

Examples

1q5.0 q5b q5.1_explain q5.2 Score Comment

I was scared and nervous. I usually got theproblem wrong if I attempted it at all.

No I think that not showing them how to do it willscare them and maybe in the long run scarethem away from math.

Not asked 0 Children will not besuccessful when asked tosolve new problems; suchexperience will bedetrimental to them.

2 (Scored re Rubric Detail 1a)q5.0 q5b q5.1_explain q5.2 Score Comment

I tried my best and usually could figure it out. Yes I would ask them to try their best and notnecessarily answer it correctly but to the best oftheir ability. It can help see what a child thinksand then address any problems specifically theymay be having.

Not very often 1 Doubt in children’s abilitiesto solve problems isexpressed.

3 (Scored re Rubric Detail 1b)q5.0 q5b q5.1_explain q5.2 Score Comment

Confussion is probably my first reaction but Ithink it is good for us to try things first on ourown to try and figure it out.

Yes It gets their brain thinking … Not every timesomething isintroduced but oftenenough for them totry things on theiroun first.

1 Problem solving will beused to stimulate thinking.Faith that children candevise legitimate strategiesis not indicated.

4 (Scored re Rubric Detail 1c)q5.0 q5b q5.1_explain q5.2 Score Comment

I was scared and most of the time did notwant ot try to solve the problem without anysort of help before.

Yes This will give me a chance to see how childrenthink and if some might be on the correct path.It will help me in understanding where I will needto begin in teaching the reall way to solve theproblem.

I am not sure. 1 Only some children will beon the right track. “Realway” indicates thatchildren’s approaches willnot be legitimate ones.

B5–S5

5 (Scored re Rubric Detail 1d)q5.0 q5b q5.1_explain q5.2 Score Comment

It was a little hard, especially with the RussianPeasant Algorithm and a few others.

Yes Kids need to expand their thinking and once ina while…

maybe like twice amonth or so.

1 No doubt about children’sabilities to devisesolutions expressed, butchildren will be allowedto solve problemsinfrequently.


My reaction was scary. I was able to learnfrom the teacher’s examples, but solving byusing my own method would be difficult.I know that at first I was overwhelmed and feltthat it was unfair of them to expect us toanswer something we have never beenshown... but then I would go back and usetrial and error with ways that I already knew.

Yes I would like to see how and why my studentsused the methods that they chose to use. Ibelieve they would learn easier and get a betterunderstanding by doing this.Yes, because even though I felt it was unfair,one feels great accomplishment when theyhave done it on their own. It shows if they puttheir mind to it they can do it.

Not too often, butjust enough toenable them to thinkabout it on their own.Maybe 45%, butprobally less

2 No doubt about children’sabilities to generatesolutions to newproblems expressed.Whether children’sthinking is to be used aspart of instruction isunclear.


My first reaction is confused because I'm notsure of how to do the problem because I havenever seen it before.Panic!!!

Yes I would ask them to solve a new type ofproblem in their own way first because it allowsthem to be creative and to find the way theyunderstand. It will also teach them that thereare more ways than one to solve a problem andthat different does not mean wrong.Yes, if this was done on a regular basis, thechildren would not panic as I used to do. Also Iwould say, “who has a different way of workingthe problem?” to encourage divergent thinkingand understanding.

I think before eachnew lesson.2x per week

3 Children are capable ofdevising solutionstrategies, ANDchildren’s solutionstrategies will be centralto instruction.

B5–S5


Aq5.0 q5b q5.1_explain q5.2 Score Comment

I felt challenged because Iwas used to being told howto solve problems. But Iwas also intrigued and feltproud once I had solved iton my own.

Yes Because it gives students theopportunity to developstrategies on their own andteaches them to be independentand not always rely on theteacher to give them theanswer.

At least 50% of the time.

Bq5.0 q5b q5.1_explain q5.2 Score Comment

Frustration and confusion Yes I think it stimulates the mindand they can play with theirimagination. They may come upwith new ideas and they mightnot always work but at leastthey tried.

Once or twice a week.

Cq5.0 q5b q5.1_explain q5.2 Score Comment

I started to panic. No, Because I feel discouragedand don’t feel like participatingwhen that happens.

Not asked

Dq5.0 q5b q5.1_explain q5.2 Score Comment

At first it was frustrating forme because I was used tobeing shown how to solve aproblem. As I worked onthe problems myself, Idiscovered that I waslearning the material betterwhen I was asked to firstsolve the problem myself.

Yes This will allow the students todiscover their own way tosolve a problem. Sometimeswhen students are shown acertain way how to solve aproblem, without realizing it,they will continue to only usethe technique they weretaught. Some students areafraid to try new ways ofsolving problems in fear thatthey will be doing it wrong.

Whenever I introduce anew concept.

B5–S5

Eq5.0 q5b q5.1_explain q5.2 Score Comment

I was frustrated andoverwhelmed. I felt like afailure and disappointed.

No Children are there to learn andexpect to be taught. By notgiving them an explanation,they could becomediscouraged.

Not asked

Fq5.0 q5b q5.1_explain q5.2 Score Comment

I found this frustratingbecause I did not have agood enough understandingof numbers to be able tosolve problems on my own.

Yes it is important to challengeyoung minds. However, I wouldprobably tell the children that itis alright if they cannot solvethe problem on their own. Aslong as they try, they will dowell.

At least once a week.

Gq5.0 q5b q5.1_explain q5.2 Score Comment

it was a challeng. but i wasalways up for trying to findthings out on my own but iknow that all of my friendshated it . they didn’t wantto do it for them selves.

Yes because i wont be there for therest of their life and if aproblem comes up that is newto them they are most likleynot going to have a teachershow them the right way to dosomthing, so i would ask themto try and then ask them howthey got there answer.

evrey once in a whilemaybe at the start ofevery chapter or everyother who knows

B5–S5


Hq5.0 q5b q5.1_explain q5.2 Score Comment

It was hard, and difficult,because at times I didn'tknow what to do, and if Idid do something I usuallygot the answer wrong.

Yes I think that at times it isnecessary for a child to find away t do it on thier own, but Iwuldn't let them struggle tomuch. I would probably givethem only a couple minutes,and I wouldn't assign it forhomework, like my teachersused to.

Probably at the beginningof a new lesson, and itwould only be f or acouple of minutes.

Iq5.0 q5b q5.1_explain q5.2 Score Comment

I think i probably sat therestaring at it for a minuteand then may or may nothave tried it on my own.

no because if they try and get theright answer by using a methodthat does not work all the timethey might get stuck in apattern of trying to use thatmethod and thinking it iscorrect. instead i would showthem a varity of ways to sovethe problem

Not asked

Jq5.0 q5b q5.1_explain q5.2 Score Comment

I hated it because it washard to get started

Yes Because it get them to think ofother ways to slove it than thetraditional way. and b/ctheother wa migt be easier forthem

3 times a week

Kq5.0 q5b q5.1_explain q5.2 Score Comment

a little confused. Yes Because, it gets their mindsthinking. They will be able tobe creative and see what theywould be able to on their own.

Maybe every couple ofweeks.

B5–S5

Lq5.0 q5b q5.1_explain q5.2 Score Comment

confused, because youdon't know how, unlessshown how too. and eventhen I may not have fullyunderstood why theproblem is solved the way itwas

Yes i would like to see which kidsare thinking on a differentscale then everyone else orwould solve it differently thenthe way i was taught. and if itmade sense, then i don't knowif i could have the studentlearn it my way.

Every now and then

Mq5.0 q5b q5.1_explain q5.2 Score Comment

I wouldn't have known howto solve a problem anyother way. I had nounderstanding of strategiesexcept for algorithms. Ipropably just sat there andtried to make up somethingthat made sense.

Yes Becuase if they do this beforelearning standard algorithmsthey have an opportunity tosolve the problem in a way thatmakes sense to them. Itdoesn't limit the ways that achild can solve a problem,therefore your students have abetter understanding of math.

every time I introduce anew subject.

Nq5.0 q5b q5.1_explain q5.2 Score Comment

When I was in high school Idid not like it at all,because I had no idea ofwhat to do.

Yes As I said I did not like it at all,but now I have learned thatkids really remember more thework when they actually figureit out by themselves

probably once a week oronce every two weeks

B5–S5


Zq5.0 q5b q5.1_explain q5.2 Score Comment

I hated it because it washard to get started

Yes Because it get them to think ofother ways to solve it than thetraditional way. and b/c theother way might be easier forthem

3 times a week

Yq5.0 q5b q5.1_explain q5.2 Score Comment

My first reaction was toestimate the answer tohave some ballpark figureto strive for. I would thenprobably deal with numbersI was comfortable with like5, 10 or 2.

Yes A teacher should only giveenough guidance so that thestudents can figure it out forthemselves. After seeing howthe students approach theproblem, I can give thestudent help based on theirstrategy, and let the oneswho understand move on, notbeing held up by the others.

Often.

Xq5.0 q5b q5.1_explain q5.2 Score Comment

I didn’t think the teachershould ask such a thing,because that was her job,to teach us how to solve it.

Yes Yes, I think it is important forchildren to learn how to thinkon their own. I know myteachers now make me do it alot.

At least twice a week.

B5–S5

Wq5.0 q5b q5.1_explain q5.2 Score Comment

Initially I was stumped asto how I could figure it outwithout prior instruction, butas I looked over theproblem more, I rememberfiguring out ways to useskills from previous units tofigure out or come close tothe answer of the newproblem.

Yes Yes, I would ask them toattempt a sample problem fromour next unit to A: give me anidea of who already has someknowledge about that unit (orwho may need more help) andB: to give students a chanceto come up with a way thatworks best for them (before Iexplain another way).

Whenever I start a newunit.

Vq5.0 q5b q5.1_explain q5.2 Score Comment

Well I wasn’t too happyabout it but I tried thesimple methods that shehad already taught us. Ifthat didn’t workthen I could always askher to give me an idea ofhow to go about solving it.

Yes Yes. I want to see what theirreaction is going to be. Itwould be interesting to seewhat solving methods theycome up with, who knows,maybe they can teach me athing or two.

At the beginning of a newunit.

Uq5.0 q5b q5.1_explain q5.2 Score Comment

i tried to solve the problemusing the reasoning that ialreay knew

Yes because it teaches te studentsto trust their judgement and totake chances

often

B5–S5



A 3Rationale for having children devise their own strategies considered strong. Engage in thispractice frequently. (We inferred that having children solve problems on their own would be anintegral part of instruction.)

B 1 Doubt is expressed: “It might not always work.” The practice stimulates the mind.

C 0 Indicates that children will be unable to solve problems.

D 3Limitations of showing children procedures for solving problems recognized. Comments in 5.1indicate respondent’s first-hand experience with the power of solving problems on one’s own.

E 0 Children will be unable to solve problems.

F 1Doubt expressed in the suggestion that children will not be evaluated on ability to solve newproblems.

G 2Strong rationale for having children devise their own solutions. Confidence that they can do it.Vague answer as to frequency is taken as evidence that problem solving will not be an integral partof instruction.

B5–S5


H 1Would let children devise solutions on their own; but doubt is expressed about their abilities to doso.

I 0 Afraid that children’s approaches will be incorrect

J 3Strong rationale for having children devise their own solutions; confidence expressed in theirabilities to do so. Will have children construct approaches frequently.

K 1Problems to solve used as a means to stimulate thinking. No suggestion that valid solutionstrategies will come from the children.

L 2Strong rationale for children’s solving problems on their own; confidence in children’s abilities todevise strategies expressed. The frequency shows that children’s constructions will not be a focusof instruction.

M 3 Children’s constructions as the focus of instruction

N 2Strong rationale for having children devise their own solutions; the frequency shows that children’sconstructions will not be a focus of instruction.

Z 3 Wants children to have alternative approaches. Will have children construct approaches frequently.

B5–S5


Y 2“Give enough guidance so that the students can figure it out for themselves” is considered apositive response. Problem solving not an integral part of instruction—doubt expressed aboutsome children’s abilities to devise strategies with comment on letting others go ahead.

X 2Importance of students’ devising strategies noted but no reason given. Degree of confidence inchildren’s abilities to devise strategies is unclear. Doubt was not expressed, so respondent wasgiven the benefit of the doubt.

W 1Doubt about children’s abilities to devise strategies expressed. That respondent will explainanother way shows that the children’s strategies will not be an integral part of instruction.

V 1No commitment to having children devise strategies—expectation that children can devisestrategies unclear; “wait and see” versus clear expression of faith in children’s abilities to devisesolutions.

U 3Vague response. “Trust their judgment” indicates confidence in children’s abilities and “often”shows that this practice would be an important part of instruction.

Scoren % n %

0 60 38% 29 18%1 87 55% 94 59%2 9 6% 15 9%3 3 2% 21 13%Total 159 159

Pre Post


B5–S7


Belief 5

Children can solve problems in novel ways before being taught how to solve such problems. Children in primarygrades generally understand more mathematics and have more flexible solution strategies than adults expect.


The question to keep in mind in scoring this rubric is “Does the respondent give evidence of thinking that children arecapable of devising solutions for problems like this on their own?” Evidence for this view is first sought in Responses7.4 and 7.5, which may be sufficient to determine a score, but Responses 7.1 and 7.3 may be needed in addition.Responses interpreted as indicating a strong belief that children can devise solutions will immediately (in 7.1) includea statement that the teacher should have offered less assistance. Responses interpreted as weaker evidence of thebelief will include in 7.3 (but not in 7.1) that the child could have done more on his own.

A difficulty of coding this rubric is that the respondent may question whether the child understands his actions on theproblem. Although this is an insightful observation, it is not necessarily an indicator that the respondent thinks that thechild could devise a solution to this problem on his own.

In this part of this survey, you will watch an interview with a child.

The following problem is posed to the child:

There are 20 kids going on a field trip. Four children fit in each car. How many cars do we need to take all 20 kids on the field trip?

Click to see the video. View Video (High Speed Connection)View Video (56K Modem Connection)

7.1 Please write your reaction to the videoclip. Did anything stand out for you?

http://coe.sdsu.edu/survey/field_trip_hi.htm

http://coe.sdsu.edu/survey/field_trip_lo.htm

Video Questions (continued)

-->>>Click if you wouldlike to see the video again.

View Video (High Speed Connection)View Video (56K Modem Connection)

7.2 Identify the strengths of the teaching in this episode.

7.3 Identify the weaknesses of the teaching in this episode.






7.4. Do you think that the child could have solved the problem with less help?

7.5 Please explain your choice.



B5–S7

Rubric Scores

0. Responses scored 0 do not indicate that the child could solve this problem with less help. Various reasons may beoffered for the child’s need for help. Some respondents state that a child will have to be taught how to solve this kind ofproblem and cannot devise a solution without specific guidance from the teacher. They show no evidence of believing thatchildren can devise solution strategies on their own.

1. Responses scored 1 show minimal evidence that the child could do more thinking for himself. The response mayinclude a suggestion that the child could solve the problem with less help, but this suggestion is not backed up withspecific information on how he might do so. Although the respondent may mention that the teacher could have “backedoff” a bit, this message is not consistent throughout the response.

2. Responses scored 2 include appropriate suggestions for enabling the child to do more of the work in solving thisproblem on his own; suggestions include to provide him with blocks and discuss the problem so he understands it. Theview that children can generate solutions on their own is expressed not in initial reactions to the video clip (7.1, as neededto score 3) but only in response to a question about teaching weaknesses or about whether the child could solve theproblem with less help.

3. Responses scored 3 provide evidence throughout all five parts to this segment that the child may have been capable ofgenerating some of the solution himself. In their initial reactions to the clip, respondents mention that the teacher tookover too much, and in their comments about the child’s ability to solve the problem, they make suggestions for how theteacher could have supported the child in thinking independently on this problem.

B5–S7

Scoring Summary


0

A. 7.4 No/ Probably Not7.5 This problem is too hard for children

B. 7.4 No/ Probably Not7.5 Children will have to be walked through this problem the first time.

C. 7.4 No/ Probably Not7.5 The child needed all the help he could get.

1

A. 7.4 Yes or Probably7.5 Does not explain how he could have done it on his own.7.3 May mention that child might have been able to do it more independently.(Respondents answer Yes in 7.4 but give no further evidence that they think the child might be ableto solve the problem on his own. We interpreted this type as the “Well, if you asked, then you mustthink so, so I’ll say Yes” response.)

B. 7.4 No/ Probably NotSome mention in 7.1–7.3 that child may have been able to do more on his own.(This is a “wishy-washy” response, indicating one thing in one place and something contradictory inanother.)

2

• 7.4 Probably/Yes7.5 Includes specifics about which parts of the problem the child might need help with and whichparts the teacher should let him do on his own. For example, “The blocks might be enough to gethim started,” or “The teacher needs to help the child understand the question.”7.1 Does not mention that child might have been able to do it more independently7.3 May mention that child might have been able to do it more independently

3

• 7.4 Probably/Yes7.5 The blocks might be enough to get him started. Or the teacher needs to help the childunderstand the question. Includes specifics about which parts of the problem the child might needhelp with and which parts the teacher should let him do on his own.7.1 Child might have been able to do more on his own if the teacher had not led him so much.

B5–S7

Comments on Scoring

0 – 1 DistinctionIf the respondent answers 7.4 No, she scores 0, unless in 7.1–7.3 she gives some indication that the child may havebeen able to do more on his own.

1 – 2 DistinctionThe respondent must respond Probably/Yes to 7.4 to score 2. She might, however, score 1 with this response if sheanswers YES but gives no further evidence that she thinks the child might be able to solve the problem on his own—the “Well, if you asked, then you must think so, so I’ll say Yes” response. To score 2, the respondent must provideideas on how the child will be able to solve the problem with less help.

2 – 3 DistinctionThe distinction between scores of 2 and 3 is related to evidence of the strength of the belief. To be considered toindicate that the belief is strong (scored 3), the response will include immediately (7.1) that the child may have beenable to do more thinking independently and give some indication in 7.1 that the child did not have an opportunity to thinkfor himself. A respondent’s raising this issue only later, in 7.3 or 7.5, is interpreted as indicating a less strong belief;such a response is scored 2.

B5–S7

Examples

1 (Scored re Rubric Detail 0A)q7.1 qvideo_7.2_reaction qvideo_7.3

_reactionq7.4 q7.5 Score Comment

I think this clip showed that kids dobetter when they have a visual tolook at.

the teacher was very reasurring tothe child. That made him morecomfortable in case he was wrong.the cubes were helpful so he couldpicture the kids in the cars.

I didn'treally seeanyweaknessin theteachingstyle

Probably not I think thisproblem wasvery hardfor a childthat youngto solve

0 The problem is verydifficult for children of thisage.

2 (Scored re Rubric Detail 0B)q7.1 qvideo_7.2_reaction qvideo_7.3_

reactionq7.4 q7.5 Score Comment

When he was first asked to solvethe problem, it seemed like he mighthave been a bit under pressure withthe camera pointing right at him. Buthe turned out to be a very goodcounter, and by teaching himvisually, he was able to solve theproblem. I commend the teacher fornot telling him his first answer waswrong, but for asking him how he gothis answer and then proceeding toshow him the right way to solve theproblem. I also like how she told himwhat type of problem he had justsolved.

Like I said before, the teacher didnot tell the student he was wrong,but instead asked how he got hisanswer and then showed him theright way to solve the problem.She also encouraged him by tellinghim what a good counter he was,and told him exactly what type ofproblem he had just solved.

I did notdetect anyweaknesses.

Probably not It seemedthat thechild didn'thave anypriorknowledgeof divisionproblems,so I thinkhelp wasneeded.

0 Children would need priorknowledge of division tosolve this problem. Therespondent indicates thatthe child needs to be shownby the teacher how to solvethis problem.

3 (Scored re Rubric Detail 0C)q7.1 qvideo_7.2_reaction qvideo_7.3_r

eactionq7.4 q7.5 Score Comment

When the teacher had the child usemanipulatives, he was able to seehow many children would actually fitinto each car, as opposed to before,he just made a guess.

Manipulatives were used. Also theteacher didn't immediately correcthim, she let him work it out on hisown.

maybe themanipulativescould havebeen shapedlike cars.

Probablynot

He needed to beguided to thinkthe way heneeded to thinkto successfullysolve theproblem.

0 No indication given that thechild might have been able tosolve the problem with lesshelp. The response in 7.3indicates that the childneeded even more help thanhe was given.

B5–S7

4 (Scored re Rubric Detail 1A)q7.1 qvideo_7.2_reaction qvideo_7.3


I think it is very typical for a child tojust guess at the answer the firsttime, but when he used themanipulatives to solve the problemthen he can get the answer right.This would instigate a child's mindthinking and allow him to feel betterabout himself while doing math.

The teacher could scaffold thechild to get the answer correctly bygiving him manipulatives to solvethe problem, by telling him how wellhe was doing, and also by tellinghim he has just solved a mathproblem that is very hard for hisage.

She didn'twait longenough forthe child toanswereachquestion.

Probably I think givenmore timethe childwould getthe answercorreclybecause hehad the rightidea.

1 The child could have solvedthe problem if the teacherhad waited. The respondenthas some idea that childrenmight be able to do more ontheir own. She expressesno clear idea of what wouldbe required for the child tobe able to complete theproblem.

5 (Scored re Rubric Detail 1B)q7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I thought the counting blocks wasa great way to bring the problem tolife and let the child visually seewhat was happening. I like howthe teacher first let him guess,then helped him see what thegroups represented when hecounted them out.

The teacher gave positivefeedback to the studentabout being a goodcounter and was also veryhelpfull in letting the childsee what the groupsrepresented.

What I like alsocontradicts what I din'tlike. The child probablycouldn't have figured outwhat he was solving hadhe needed to do it onhis own. The teacherseperated the blocks forhim and labeled eachgroup instead of letteinghim do it. At his age itmay be necessarythough, but he mayhave a little trouble if hewas asked to do similarproblems without anyguidance.

Probably not I don't thinkhe wouldhave knowwhat eachof thegroups weremeant torepresent,but maybehe wouldhave figuredit out.

1 This respondent providescontradictory statements.She indicates (q7.3) that theteacher could have backedoff a bit. She suggests(q7.5) that he might havebeen able to figure out someon his own, but sheexpresses great deal ofdoubt. The response wasinterpreted as being “wishy-washy,” indicating weakevidence of the belief, so itwas scored 1.

B5–S7

6q7.1 qvideo_7.2_reaction qvideo_7.3


Without manipulatives, the child wascompletely unable to visualize thereality of the problem. Once hecould pretend like the blocks werethe 20 children, he was able to seethat 20 kids divided into groups offour would require the use of 5 vans.I like the way the teacher guided hislearning with specific questioning.

The teacher's use of questioning. maybegiving alittle toomuchinformation

Yes I think theteachercould havestopped bytelling him toget out 20blocks torepresentpeople. atthat point,the childcould havemade thegroups onhis own andcome upwith his ownconclusion

2 The respondent has a clearidea that the child couldhave solved the problemwith less prompting. Shesuggests that if the teacherencouraged the child to usemanipulatives to beginacting out the problem, heprobably would have beenable to continue on his own.This response is not scored3 because the respondentdoes not indicate in 7.1 thatthe child would have beensuccessful with less helpfrom the teacher.

7q7.1 qvideo_7.2_reaction qvideo_7.3


This was a nice introduction on howto solve the problem. I like the useof the manipulaives, but I think thatthe interviewer did too much of thework for the child.

The teacher was very patient andclear with the child.

The teachergave awaytoo much ofthe answer.She couldhave ledhim toansweranything.She didntallow him tothinkenough onhis own.

Probably I think thatwith time,the childwould havecome to thecorrectanswer. Hedefinatelyneededsome help,likerepresentingthe kids withblocks, but Ithink that hecould havetaken it fromthere.

3 The respondent mentions(q7.1), even before beingprompted about weaknessesin the teaching, that thechild was coached toomuch, interpreted asindicating that the belief isstrongly held. Therespondent has an idea forhow the teacher couldsupport the child’s thinkingwithout doing the thinkingfor him.

B5–S7


Aq7.1 qvideo_7.2_reaction qvideo_7.3_reactio

nq7.4 q7.5 Score Comment

I think that is was neatthat when she had himuse the manipulatives tosolve this problem hesaid, "this is easy." Heneeded some guidance tosolve this problembecause without it he justguessed.

The strengths were that sheused a lot of goodquestioning. She didn't givehim the answers but shehelped him understand whatshe was asking.

When she firstasked him theproblem she gaveno guidance.

Probablynot

Because he quicklyguessed and showed nointerest in trying to figureout if he was right orwrong. He seemedhappy that he'd come upwith an answer and ifsomeone hadn't beenthere to help him heprobably would have lefthis answer as 20.

Bq7.1 qvideo_7.2_reaction qvideo_7.3_reactio


Having done this problemwith a student in the past itwas interesting to see howmuch a like the methodswere. The student took astab in the dark at first butwhen they worked it out alittle bit with the blocksthey understood it more.

The strengths were: Nottelling the child exactlywhat to do rather guide himalong and also the positiveencouragement along theway helped him with hisconfidence in doing theproblem.

I don't think thatthere were any.

Probably I think that he couldhave done it with lesshelp, but it would havetaken him a long time.

Cq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I noticed the manipulativeswere used to help the childwork through the problem. Inoticed that the teacherempowered the child byhaving him demonstratedcounting in the process ofsolving the problem. Inoticed the specificquestions that were asked toget the desired responsefrom the child.

Questioning, use ofmanipulatives,beginning the processon a level that the childcould understand anddo (counting), clearquestioning, etc.

I did not observe anyweaknesses.

Probablynot

The child did notseem to have beentaught how to workout the problem inhis mind nor withmanipulatives.

B5–S7

Dq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

He did everything thelady asked him to do, butI'm not sure heunderstands what he did.Counting out the cubesand breaking them intogroups of four, was alldone by her command.Otherwise, I feel it was agood strategy to showthe boy how to solve theproblem. Using cubesand dividing them intogroups is very effective.

Lots of positive reinforcementand energy from the teacher.She asked him to do all thecounting freely. Explained atthe end that what he haddone was division. Sheasked him to try to solve theproblem himself before shehelped him.

She never asked himfor feedback to see ifhe understood whatwas happening withthe groups of cubes.

Probably She couldhave givenlessinstruction andhe probablywould havebeen able toanswer. Andthis would bebetter for thechild.

Eq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

The one thing that stoodout to me was theteacher leading the childtoo much. I think theteacher did not giveenough time to the childso he could solve ithimself. I think that ifshe had helped him lessand just slightly steeredthe student he wouldhave got a betterunderstanding at the endthe student did not seemto understand.

Positive reinforcement Too much leading bythe teacher

Yes I think thechild wouldhave solvedthe problem ifhe had moretime and if theteacher didnot interfereso much.

B5–S7

Fq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I liked how the teacherlet the student figure outthe answer on his owninstead of at thebeginning just telling himhe was wrong. Sheasked how he got hisanswer, then let him usethe blocks to figure outthe problem. She guidedhim through by askingquestions! Now he feltsuccessful because hefigured it out himself.

The questioning was agreat way to guide thestudent through theprocess. The teacher alsomade him feel like he wasreally good because hefigured out a divisionproblem.

The teacher couldhave had the studentput the blocks in"cars" instead ofgroups, then cars.She could haveeliminated someconfusion by skippingthe group idea andgoing straight to thecar idea.

Probably The child guessedat the beginning outof laziness as faras I could tell.Maybe he couldhave figured outhow to put theblocks in "cars" bythe teacher justasking him toinstead of leadinghim through eachcar.

Gq7.1 qvideo_7.2_reaction qvideo_7.3_reactio


When given the problemat first the student wasoverwhelmed and justguessed the answer. Hedidn’t think he knew howto solve it. But when heused the blocks tosymbolize the studentshe was able to solve theproblem.

The teacher talked the childthrough the exercise. Shewas very encouraging to thestudent.

I feel that shehelped the child outa little too much. Ithink he would havebeen able to solvethe problem with theblocks without herhelp. He justneeded to visualizethe numbers.

Yes The child justneeded to see thechildren visually.He knew how toseparate the blocksinto 4 groups. He’slearned subtractionin class so he wouldbe able to userepeatedsubtraction to solvethe problem. Hejust needed a littlebit of time.

Hq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

The student could answerthe problem because hecould use manipulativesto solve it and he hadteacher scaffolding.

The teacher was good athelping the boy think throughthe problem logically.

I don't know if therewere any.

No It seemed thatthe studentneeded thehelp he got.

B5–S7


Iq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I noticed how the kidpicked out the colors heliked when he wascounting them. And heseemed to be a littleconfused until the end.He know he solved theproblem but he didn'tseem to understand why.

she never told him he waswrong and he wasencouraging

She didn't elaboratemore on at the end. Icould tell that he wasconfused and itseemed like once shestarted counting outthe cars for him thenhe just continuedcounting withoutunderstanding why

Probably he might haveif he wasgiven a littlemore time andguidance

Jq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I was surprised to seethat the student chose totake a guess rather thantrying to solve theproblem or admitting thathe didn't know how.When he thought about itfirst I thought that hewas trying to visualize it.He seemed to be proudwhen the problem wasbroken into parts that heconsidered easy. Whenhe counted out thegroups of four he stilldidn't realize what thosegroups represented untilthe teacher got himstarted counting out the"cars"

The teacher put it into partsthat he could easily do so hefelt like he had accomplishedsomething and even whenshe had to help him a lot shetold him his strengths and lethim give the final answer.

She had to do toomuch work for him andmaybe should havegiven him more time tolet him think after shegot him started. heprobably could havefigured out what thegroups of fourrepresented or that heneeded to keepcounting out groups offour after he had thefirst one done.

Probably Once he gotstarted andthe teacherhelped himmake one"car" heprobably couldhave figuredout that heneeded tokeep countingout fours tomake more"cars" andcould havefigured out tocount up thetotal numberof cars at theend.

B5–S7

Kq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

The manipulatives workedgreat in this problem.When the child wasasked to solve theproblem, he guessed.With a little coaxing itwas easy for the child tosee that four kids fit ineach car.

the interviewer coaxed him,by asking him to show herthe 20 cars. Then asked tosee groups of 4. He evenstated that the group of fourwas easy. She didn't tellhim the answer but helpedhim to understand what hewas grouping and he wasable to see and solve theproblem.

I don't think there wasany

Probably not He seemedthat he didn'tknow how tosolve theproblembecause heguessed atfirst. he alsoseemed to bein the first orsecond grade,so I don'tthink that hehad solvedmany divisionproblemspreviously.

Lq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

umm.... i think theinterviewer did a great jobof solving this problem...

well, manipulatives wereused... that's good... thechild was able to see how aproblem could berepresented with theblocks... maybe he hadn'tsee that before

i think it would've beenmore beneficial if thechild did it for himselfinstead of justwatching theinterviewer do thewhole problem. like ifshe has said, "so howmany kids can fit in acar? how could weshow that?" then thelittle boy would have tothink about it, he wouldbe connecting thewords with themanipulatives forhimself... yeah... ithink it would've been astronger lesson if thathad happened andlikewise with the nextsteps

Yes he might'veneeded a littlehelp, but notas much aswas given tohim

B5–S7

Mq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

The video showed aperfect example of howthe visual aids can help astudent learn the correctanswer. I like how theteacher was so positivewith the student. Eventhough his guess waswrong in the beginning,she still reinforced thathe was a good counter

The teacher was a positivereinforcer that wholesession. She verbally saidthe problem and then whenthe student guessed wrong,she didn't tell him that hewas wrong, but had him usethe blocks, which is good forvisual learning. The wholetime she gave him theconfidence that he was agood counter and in the end,she made him feel smart,like the "big kids."

In the end she shouldhave let him count thecars, but I understandthat she had to start himoff. I think that shehave told him at the endthat his guess wasclose, but that the realanswer was five.

Probablynot

To me thechild wasconfused anddidn't knowexactly how todo theproblem. Thevisual aidswere a greathelp, but shehelped him alittle to much.

Nq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

Using the blocks was agood stragedy because itshows a 'big picture' onhow to solve the problem.I think using tools isgreat for the studentsbecause at the beginningof the clip, he tried tothink about how to solvethe problem, but hecouldn't. When he usedthe blocks, he was ableto solve the problem. AsI said before, usingobjects is an easier wayfor children to see theproblem.

She was able to help himstart the problem by tellinghim to count the 20students, and that there are4 students per car.

She was just telling himwhat to do, not lettinghim think about what todo next on his own.She should have let himthink about it first,rather than telling himwhat to do right aftercounting the blocks.

Probablynot

Right at thebeginning, Icould tell thathe was goingto need help insolving theproblem. Itseemed as hecounted the20 blocks,which werestudents, hehad no idea onwhat to donext. He wasconfused. Ifshe explainedit better, thenhe probablywould havesolved thproblem on hisown, but hewould stillneed a lot ofhelp.

B5–S7


Zq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I liked the way sheworked through eachstep of the problem withhim. The step by stepmethod of having himshow 20 students thenrepresenting 1 car with 4students seemed toreally help, even thoughhe didn’t fully understandthe concept he wasworking with.

The teacher made sure heunderstood all elements ofthe problem, like theamount of students in totaland how many each carcould hold. It also allowsthe student to work withmanipulatives, which Ibelieve help tremendously.

There was a lot ofprompting by theteacher.

Probably If the problem were set upthe same way and thestudent was allowed towork through the problemby exploring the relationbetween the numbers. Hemay have gotten it wrongand taken more time, but Ithink he would haveeventually gotten thecorrect answer.

Yq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

Many times students don'tthink about the mathproblem because theythink it will be too hard, sothey guess, and most ofthe time the guess is wayoff. I have found that itmath is so much easier todo math if you havevisuals; it makes things somuch easier to see andthe next time you canvisualize it in your head.

Having the student answerbefore using the visualaids let's the teacher gagehow far along the studentis. Also having thestudents think about it andthen use the visualsmakes the problem andmath in general so mucheasier than it seems.

It takes time andworking withstudents individuallyor in small groups.

Probably I don't think that thisstudent could translatewhat the teacher wassaying into a mathproblem. If he wasgiven the problem in adifferent form, maybepicture form, heprobably could havesolved it on his own.

Xq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

The teacher stepped inwith hands-on and hadthe student figure it outthat way. Since thestudent guessed, it wasimportant for the teacherto show a way theproblem could be done.

She recognized that sheneeded to step and helpthe students. Sheshowed her step by stephow to do it.

She could have hadthe student explainhow each step washappening as she wentalong. The studentseems to be seeing ithappen, but not reallyunderstanding._

Probablynot

I think that if the childplayed around with theproblem, she could havefigured it out. However,it was important that theteacher jumped inbecause the student maynever have gotten it.

B5–S7

Wq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I liked how the teacher didn't settlefor the child's estimated guess.However, I was surprised that shedidn't offer the blocks for him to useright off the bat. Maybe he wouldhave used them instead of randomlyguessing right away. I also found itinteresting that she didn't ask him hewould try to find the answer using theblocks on his own first. She kind ofjust walked him through the problem.

She didn't settle forthe child's randomguess as his answer.

She walked himthrough the problemand it seemed asthough she did histhinking for him. Atthe end of theproblem, I'm not sureif he quiteunderstood why theanswer was 5 cars.

Probably I think that the childcould have solvedthis problem withless help from theteacher. I believethat if he used theblocks on his own,he could haveeventually solved it.

Vq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

The student could answer theproblem because he could usemanipulatives to solve it and he hadteacher scaffolding.

The teacher was goodat helping the boythink through theproblem logically.

I don't know if therewere any.

No It seemed that thestudent needed thehelp he got.

Uq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

I thought it was interesting that theboy first just guessed and the teacherasked and acknowlegded that it was aguess, but then went on from there.She started with something the boycould do count. She then broke itdown in to a series of steps the childcould grasp. then at the end shecongratulated the child on being ableto do division. Something I am surehe thought he could not do. I loved itbecause the child saw what and howto do the steps without having to knowthat he was doing a scary problem,just solving a simple everydayproblem. It was a real concern.

guiding the childgiving support andasking about thechild's thoughtprocesses. Havingthe child think a loud.

The child needstime to continueworking on thesetype of problemand be able toworkindependently.

Probablynot

I think it was the firsttime he had thoughtabout this type ofproblem. I think if hewas to do it againand be allowed toexplain why he didwhat he did it wouldbe beneficial

Tq7.1 qvideo_7.2_reaction qvideo_7.3_reaction q7.4 q7.5 Score Comment

i thought that it was a veryeffective way to introduce astudents to division. He wasable to see what he wasdoing

she let him figure it outhands on, she had a goodexplanation and clearobjective. She gave himgood immediate assessment

what was going on withthe reset of the class?How do you accomplishsomething like that in awhole class setting

Probablynot

He seemed soconfused with theproblem that he wasjust guessing at thefirst

B5–S7



A 0Child needed more help initially to solve this problem; indicates belief that children cannotdevise solutions for themselves.

B 1Perhaps the child could have done more on his own (7.5), but no specifics offered aboutwhich aspects of the problem he could have done independently. (Type 1a response)

C 0 The child needs to be taught how to do this problem before he will be successful.

D 1

This response is difficult to code because the respondent notes the child’s lack ofunderstanding in 7.1. This insightful observation is easily mistaken for, but is not thesame as, an observation about the child’s ability to solve the problem on his own. Thisrespondent is concerned about the child’s understanding in both 7.1 and 7.3, but only in7.5 does she suggest that he could do more of the thinking on his own, and then sheprovides no specifics about what the child might be able to do on his own.

E 3The teacher was leading the child too much (7.1). Indicates a strong belief that childrencan devise solutions on their own by recognizing from the first that the child was not giventhe opportunity to do so.

F 1States that the child might have been able to solve this problem with a little less help butindicates that he needed most of the help, except for the teacher’s directing him to createeach group of four.

G 2The child could have solved the problem if the teacher had given him the blocks andallowed him to work on his own. Respondent does not note in 7.1 that the child could havedone more on his own.

B5–S7


H 0 The child needed all the help he received.

I 1 The child could solve the problem his own; no indication given of how he might do so.

J 2 The child could have solved the problem on his own (but mentioned first in 7.3, not 7.1).

K 0 The child could not have solved this problem on his own.

L 3 The child could have done more on his own (indicated in 7.1).

M 1Although the child could not solve this problem on his own, the teacher may have helpedtoo much. (Given benefit of the doubt and scored 1).

N 1Inconsistent response. Response 7.3 indicates that the child might have been able to domore on his own, but 7.5 is unclear with respect to whether the child could solve theproblem on his own or needed a lot of help. (Type 1b response)

Z 2

This ambiguous response could be scored 1 or 2. The explanation for how the child mighthave been able to solve the problem on his own does not specifically include blocks orhelping the child understand the questions but does include the notion that exploring thenumber relations would allow him to solve the problem.

B5–S7


Y 2The child will be able to solve this problem if it is presented in a more comprehensibleway.

X 1The child probably could not solve this problem on his own, but he might have been able tohad he been given the opportunity to “play around” with the problem.

W 3The interviewer walked the child through the problem rather than giving him the blocks andhaving him work the problem out on his own (stated in 7.1 and reiterated in 7.3). The childprobably could have figured out what to do with the blocks on his own (7.5).

V 0 The child needed all the help he got.

U 1This response could be scored 0 or a 1. We gave the respondent the benefit of the doubtbecause she said that the child needed time to continue working on these problems.

T 0 This is a difficult problem, which the child needed help to solve.

Scoren % n %

0 75 47% 71 45%1 42 26% 43 27%2 41 26% 35 22%3 1 1% 10 6%Total 159 159

Pre Post


B6–S2


Belief 6

The ways children think about mathematics are generally different from the ways adults would expect them to thinkabout mathematics. For example, real-world contexts support children’s initial thinking whereas symbols do not.


Coders should focus on whether responses indicate sensitivity to children’s thinking, in particular, whether therespondents recognize that young children typically solve the first problem using manipulatives to add on and can solvethe second problem by making three sets of five. Respondents who show insensitivity to children’s thinking in oneaspect of their responses and sensitivity in other aspects are considered to be providing weak evidence that they holdthis belief whereas respondents who show sensitivity to children’s thinking throughout are considered to be providingstrong evidence of holding this belief.

In considering responses to Items 2.2 and 2.4, coders should keep in mind that the wording of these items includes“convincing a friend” about one’s response. Reliability issues arose when coders were learning to distinguishbetween 0 and 1 scores. Later reliability issues arose in their distinguishing between scores of 1 and 2 for ambiguousresponses.

Note. Coders developed a way of talking about the responses. They noted when they saw a glimmer of sensitivity tochildren’s thinking (for example, noting that the first problem was NOT a simple subtraction problem was a glimmer)and when they saw a negative, which was insensitivity to children’s thinking (“children will write an equation” wasconsidered a negative).

Note that this rubric does not assess the respondent’s holding of Belief 5 (concerning children’s abilities to devisesolution strategies on their own) or of Belief 7 (concerning the amount of guidance provided by the teacher). The focusof this rubric is on whether the respondent recognizes that children’s interpretations of mathematics differ from adults’interpretations. Some respondents discuss teaching approaches that are consistent with children’s thinking. Theirscores are high on this rubric but low score on the rubric for this segment in assessing Belief 5. Other respondentsexplicitly describe how children would interpret the problem but doubt they will be able to solve it. Such responses alsoreceive a high score on this rubric but a low score on the rubric for assessing Belief 5.




Yes

No

B6–S2

Rubric Scores

0. Responses scored 0 indicate that children will approach both problems in the ways that adults do: Children will interpretthe first problem as subtraction and the second problem as multiplication. Respondents may state that children will needto convert the problems into equations before they can solve them. They do not recognize that children might act out thestory problems in accordance with the actions in the problems. They show no sensitivity to children’s thinking in the waysthey respond to this question.

1. Responses scored 1 indicate sensitivity to children’s thinking in one minor respect but insensitivity otherwise.Respondents may, for example, recognize that the Pokemon problem is not a simple subtraction problem but fail torecognize that children could model this situation according to the action in the problem. Others mention that the contextor a visual representation will help children to solve the problem but make other comments that indicate insensitivity tochildren’s thinking.

2. Responses scored 2 indicate sensitivity to children’s thinking for one of the problems but not the other. Respondents maynote that the Pokemon problem is not a simple subtraction problem and that children could solve it by adding on in someway but then go on to note that children cannot solve multiplication because they have not been formally introduced to it.Others receiving this score give a detailed explanation of how children might solve one of the problems but not the other ornote, without providing details, that children can solve both problems by using manipulatives. Respondents give someevidence of understanding the power of children’s approaches, but the evidence is insufficient to convince us that thebelief is fully developed.

3. Responses scored 3 show sensitivity to children’s thinking throughout, either through a detailed explanation of howchildren might solve both problems or a detailed account of how a familiar context and the opportunity to model will enablethe child to solve the problems.

B6–S2

Scoring Summary


0 • Children can approach the problems only in the ways adults do. The first problem will be solved assubtraction and the second problem will be solved as multiplication.

1A. Join Change Unknown problem is NOT simple subtraction; recognition that children could act out the

problem not included.B. Visuals mentioned OR appreciation that the context will support children’s thinking stated

BUT insensitivity to the ways that children naturally interpret problems shown in the response.

2

A. Detailed explanation for how the child will solve the missing-addend problem but not the multiplication;may state that the child cannot solve the multiplication.

B. Detailed explanation for how the child will solve one problem but vague explanation for how child willsolve the other

C. Child will use cubes/fingers (detail lacking); answer shows sensitivity to the ways that children naturallyinterpret problems.

3• Detailed explanation of how children might solve each problem

All aspects of answer show sensitivity to children’s thinking—indicates that modeling story problems isa natural approach for children when the context is familiar.

B6–S2

Examples

1q2.1 q2.2 q2.3 q2.4 Score Commentyes I have used questions like this with my

first graders at work and they need alittle help at first but most can realize thequestion is a subtraction

no It is harder for a first grader to domultiplication.

0 Pokemon problem will be done assubtraction and the multiplication will beout of reach of young children.Sensitivity to the ways that childrenapproach mathematics not shown.

2 (Scored re Rubric Detail 1A)q2.1 q2.2 q2.3 q2.4 Score Commentno I think that this problem would be very

difficult for a first grader. When I workedin a first grade class word problemsseemed to be the hardest. This is asubtraction word problem that I believewould be too difficult. The kind ofsubtraction word problem that I think afirst grader could understand is: Mariaowned 13 Pokemon cards, she lost 8pokemon cards, how many cards doesshe have left.

no This is a multiplication word problem. 6and 7 year olds have not learnedmultiplication yet, or repeated addition.

1 The Pokemon problem is not a simplesubtraction problem (shows sensitivityto children’s thinking). Otherwiseresponse lacks sensitivity to children’sthinking in indicating that the Pokemonproblem must be solved as asubtraction problem and that childrenmust be formally introduced tomultiplication to solve the gum problem.

3 (Scored re Rubric Detail 1B)q2.1 q2.2 q2.3 q2.4 Score Commentyes After having personal experience with

children of that age group, I know kids ofthat age group are able to comprehensuch a problem. A child by that time canusually count quite well, and much of thetime you will see the child arrive at theiranswers by counting on their fingers.

no This problem I believe has a little toomuch information for a fisrt grader to takein or process. This problem could alsobe solved with multiplication, andanything with that degree of difficulty fora first grader is quite advanced. At thispoint in time they are just learning how togroup objects in numbers, and to put aproblem such as this into a word problem,and not visiual I believe is too advancedfor that stage.

1 Sensitivity to children’s thinking shownin statement that they will count on theirfingers and need visuals. Details notprovided for how children would solvethe first problem. Comment thatmultiplication problem is too hard forchildren because it is a word problemshows insensitivity to children’sthinking.

B6–S2

4 (Scored re Rubric Detail 2A)q2.1 q2.2 q2.3 q2.4 Score Commentyes I would say that a child could solve this

problem but not the way it is usuallysolved (subtraction). The child could addmore cards until she reached 13. Thenthe child could go back to see how muchshe added.

no A child in this grade cannot grasp theconcept of repeated addition.

2 Details given for how the child will solvethe Pokemon problem, showingsensitivity to children’s thinking.Sensitivity not shown in gum-problemresponse.

5 (Scored re Rubric Detail 2B)q2.1 q2.2 q2.3 q2.4 Score Commentyes This is a missing addend. The child has

the starting numbr and the finishingnumber. They can count up using theirfingers, manipulatives or in their head. Ithink with these methods they can figureout the problem.

yes Children are often effective in solving thiswith manipulatives. Often I think wedon't give children enough credit for whatthey can think about and solve.

2 Details given for how the child will solvethe Pokemon problem but explanationfor the second problem is vague.

6 (Scored re Rubric Detail 2C)q2.1 q2.2 q2.3 q2.4 Score Commentyes I would say that if you read the problem

to the child and had visual aids to showwhat the problem is asking then the childmight be able to see what was going onand answer the problem with the cardsthat you set out.

yes I think that with the teachers help thestudent could solve this problem. But afirst grader could not do this on his/herown. The teacher or helper would haveto guide them through the step forsolving the problem. They would read theproblem, get out 3

2 Even though the respondent believesthat children need to be taught thesestrategies, the respondent mentions theimportance of visual aids and iscognizant of the ways that childrenapproach these problems. Because ofthe lack of detail for solving the JoinChange Unknown problem, the score is2, not 3.

7q2.1 q2.2 q2.3 q2.4 Score Commentyes A first grader would be able to see that

13 is more than eight and using theirfingers, or some other implement theycould count from eight (nine, ten, eleven,twelve, thirteen) and see the difference isfive. It is the same as subtraction butusing the addition they already know.

yes If the child has learned sorting andgrouping it is a very easy problem. Theywould separate the gum packs (usingblocks or cubes) into three groups,making sure that each group has 5 sticksand count them all. It is multiplication,but once again is using the addition theyalready know.

3 Both responses give detailedinformation of how children could solveboth problems by using manipulatives tofollow the action in the problem.

B6–S2


Aq2.1 q2.2 q2.3 q2.4 Score Commentyes you can lay them out on a

table and show the firstgrader that there are thirteenthere and if Maria alreadyhad eight, then you wouldtake 8 cards away from thetable and have the firstgrader count the remainingcards on the table and showhow this work

no i do not think that a firstgrader would be able tocomprehend this type ofmath unless visual aideswere used by laying outthe gum...... thisproblem is too complexfor the age.

Bq2.1 q2.2 q2.3 q2.4 Score Commentno It is basic algebra, and

unless they are taught thetheory behind algebra, theywill be confused. Maybe afirst grader could figure itout, but it would take a lot ofthinking and a lot of hands-on.

no This is multiplication andgenerally students donot learn this until thirdgrade. I also think thatit is a very difficultproblem to a first graderand that they could noteven be able tounderstand the problem,let alone, solve it.

Cq2.1 q2.2 q2.3 q2.4 Score Commentyes It is my understanding that

by first grade the studentshave already learned theirnumbers, so with thisknowledge they have aquiredthey can begin tounderstand that if you have8 cards if you add another 5cards that gives you a totalof 13 cards. It i

yes I would again explainthat this is a problemthat if the student has ahard timecomprehending can beexplained visually bygrouping the 3 packs ofgum individually andthen adding up howmany sticks of gumthere are in total.

B6–S2

Dq2.1 q2.2 q2.3 q2.4 Score Commentno That first graders do not

understand the use ofaddition and subtraction in aword problem

no First graders have littleor no experience withmultiplication

Eq2.1 q2.2 q2.3 q2.4 Score Commentyes I think that this type of

simple addition in the form ofevery day life examples thata first grader would knowhow to solve. They couldcount on their fingers from 8to 13 as well as usingblocks, cubes, beans, oreven drawing it out on asheet of paper

no I don't think first gradershave learned to domultiplication yet. Evenif they just made piles ofcubes with three piles offive cubes each, i stillthink it is a little bitabove their level.

Fq2.1 q2.2 q2.3 q2.4 Score Commentyes you could bring cards into

the class and demonstrateit. you could also set up theproblem as 8 plus whatequals 13. this problemactually might be a littletough for a first grader.

no i don't remember doingmultiplication in firstgrade and that is whatthis problem involves5x3= 15. i think firstgraders can only handleadding and subtracting

B6–S2


Gq2.1 q2.2 q2.3 q2.4 Score Commentyes The problem is one which

might relate to the child, andI believe that by first grade Icould do simple addition andsubtraction. This problem isjust a matter of subtraction,so I think that the childwould be able to figure itout. Even if they had to

no Multiplication seems likea rather large step for afirst grader. I believe Ilearned multiplication insecond grade, but they(the child) might be ableto break it down andsolve it by adding, butthat would be a step, Ibelieve, for a firstgrader.

Hq2.1 q2.2 q2.3 q2.4 Score Commentyes its easy to see that she

started with 8 and now shehas 13 so the child wouldknow to add to 8 until theyreached 13, or just subtract8 from 13

no most first graders don'tknow how to domultiplication problems

Iq2.1 q2.2 q2.3 q2.4 Score Commentno I do not know much about

how much a first graderknows. I would tell themabout my cousin who is inkindergarten and that hedoes not know how to addyet

no This is multiplication andthat is not taught untilthe 4th or 5th grades

B6–S2

Jq2.1 q2.2 q2.3 q2.4 Score Commentno Word problems are very

confusing and depending onhow well the parents workwith the child, then he/shemay understand the problem,but I feel a typical firstgrader would not be able topull out the correct theinformation to solve theproblem

yes A child can visualize thepacks of gum and cansee how many sticksare in them. If a childdoes this then he/shecan simply count howmany sticks he/she hasand find the right answer

Kq2.1 q2.2 q2.3 q2.4 Score Commentyes children often like to count

up to find a difference, achild could easily say "leticiahas 8 cards at first, andthen she has (counting onher fingers up to five) nine,ten, eleven, twelve, thirteencards, thats five cards."

no I just spent sevenweeks in a secondgrade classroom wherethey were doing their 3times tables, which tellsme that they are justlearning how to multiplyby threes. I spent a lotof time babysitting a girlwho was in first grade,and she was only doingaddition and subtraction.If the first grader is ableto have the problem infront of them, and writetheir work out, they maybe able to do that. Thechild could draw outthree boxes with fivesticks of gum in eachbox, and then count howmany there are total,which would be addition.I think that some firstgraders could do this,but not a typical firstgrader.

B6–S2

Lq2.1 q2.2 q2.3 q2.4 Score Commentyes if a friend disagreed with me

I would tell her in letting achild use some kind of visualprop like blocks or so theywould definitely be able toanswer the problem

no i would tell them thatthey would definitelyhave difficultiesbecause they would tryto add the problemrather than multiply.

Mq2.1 q2.2 q2.3 q2.4 Score Commentno this is a join, change

unknown promblem, that maythrow the child off. bysecond grade they wouldhave no problem. some firstgrade students may noteven have a problem with it.they would most like use themethod of count up, usingtheir fingers to keep track ofthe # they added.

no once again this isslightly aove theaverage first grader.some of the studentscould probably come upwith an answer,especiallly if they hadmanipulatives to workwith. most secondgraders would be able tosolve this problem,some of them mayrequire manipulatives.

B6–S2


Zq2.1 q2.2 q2.3 q2.4 Score Commentyes I would tell them that all the

child needed to know wasthat she had 8 in herbelonging. And after herbirthday she now has 13.Thus, the average childshould be able to figure outthat the two totals weredifferent and taking thedifference of the two shouldgive you the answer, five.

no I'm not sure if firstgraders learnedmultiplication yet, buteven though, theyshould be able to figureout that adding 5 threetimes should give youthe answer. However, Idoubt the abilities of afirst grader tocompetently solve thisproblem.

Yq2.1 q2.2 q2.3 q2.4 Score Commentno I think that a typical first

grader would forget aboutthe given information andconcentrate on the lastfigure given, 13.

no This problem is fairlymore complicated indescribing the situation.I'm afraid that the childwould confuse 'packs ofgum' with 'sticks ofgum'.

B6–S2

Xq2.1 q2.2 q2.3 q2.4 Score Commentno In order to solve this

problem a child needs to usesubtraction. They need tomove the problem around ina way that they willunderstand it. I think that afirst grader could do theproblem if they had thecards in their hand and wasable to go through themotions of having 8 and thenadding five more to make 13cards.... or by have 13 andtaking 8 cards away to figureout the number of cardsneeded to make 13.

no That i see as basicmultiplication. It is 3packs of gum times 5pieces per pack whichsums up to be 15pieces. The only way afirst grader could do thisproblem is either byusing visuals or byadding five togetherthree times, but youwould have to explain tothe child that there arethree fives and why.

Wq2.1 q2.2 q2.3 q2.4 Score Commentno Well, my sister is in first

grade and she is prettyadvanced for her age and Idon't think that she woud beable to figure this out. Itseems that students havedifficulty working withmissing numbers. Now if itwas a simple, Leticia had 13cards and gave 8 away, Ithink that they would be ableto figure this out.

yes With some assistance Ithink that a child couldfigure this out.Manipulatives wouldhelp, but most 1st gradestudents would look atthis problem, not asmultiplaication, butadding 3 groups of five.

B6–S2

Vq2.1 q2.2 q2.3 q2.4 Score Commentyes i didn't really know how to

answer that questionbecause i'm not sure of theability of a "typical" firstgrader. A typical first graderat some schools i've workedat definatly would not beable to answer that and visea versa.

no i'm not sure if firstgraders have reachedthat level yet.

Uq2.1 q2.2 q2.3 q2.4 Score Commentyes This is a fairly easy problem

for a first grader. It isaddition and the numbersaren't big. A first gradercould draw a picture or usehis/her fingers to solve thisproblem.

no I would have to say thatI don't recall learningmultiplication in firstgrade and I have afourth grade brotherwhom I've helped withhomework quite oftenand I didn't seemultiplication problemsuntil the end of secondgrade.

Tq2.1 q2.2 q2.3 q2.4 Score Commentno Well, I work with k-3 graders

and have witnessed that firstgraders can basically onlyadd or subtract withpictures. So if there is nopicture for them with thisproblem then I am sure thatthe average first graderwould not understand.

no This is used as amultiplication problemusually and can be verydifficult with thirdgraders.

B6–S2



A 1

Sensitivity shown in comment that manipulatives will help children understand the Pokemonproblem. The actions described for the manipulatives do not align with the problem.Multiplication-item response lacks sensitivity to children’s thinking. One minor aspect of theresponses shows sensitivity to children’s thinking. (Score 1B)

B 0Sensitivity to children’s thinking not shown in assumption that both problems need to be solvedin a formal way.

C 3Strategies that align with children’s thinking described. (Assumption that the teacher mustsupply these strategies will affect score on a different belief.)

D 0Sensitivity to children’s thinking not shown in assumption that children will be incapable of doingeither problem.

E 2Detailed description of how children can solve the Pokemon problem; although respondent notesthat children might build a representation, she doubts they will be successful. (Score 2A)

F 1

“Cards” might help, but respondent mentions representing the problem in symbolic form. Shedoes not describe how the visuals might be used to act out the problem. One minor elementshows sensitivity to children’s thinking, but other elements (introducing the number sentence)show insensitivity to children’s thinking. (Score 1B)

G 1Children can relate to the problem, but it is subtraction. Children will be unable to solvemultiplication problem. (Score 1B)

B6–S2


H 2Details given for how children will solve the Join Change Unknown problem; children cannotsolve multiplication. (Score 2A)

I 0 Children will be unable to solve either problem.

J 2Children can solve multiplication problem (explanation given). Join Change Unknown problem istoo difficult. (Score 2B)

K 3Explanation given for how children will solve each problem, even though doubt is expressedabout the multiplication problem.

L 1 Visuals are helpful. Comments about multiplication show insensitivity to children’s thinking.

M 2Importance of using manipulatives to solve these problems is noted. Statement that problemsmay be difficult for some first graders shows sensitivity to problem type. Details not given for howchildren will solve each problem, so response does not warrant a 3. (Score 2C)

Z 1Statement that multiplication might be done as 5 three times shows slight sensitivity to children’sthinking, but comment that the Join Change Unknown problem is a subtraction problem indicatesinsensitivity to children’s thinking.

Y 0Sensitivity to children’s thinking not shown in the response that children will be unsuccessful onboth problems.

B6–S2


X 3

This strange response indicates doubt that children can solve these problems in legitimate waysbut shows recognition of the ways that children would look at these problems and gives detaileddescriptions of how children could solve each problem. Evidently the respondent realizes thatchildren’s thinking is different from adults but does not consider children’s approaches to countas solutions (this perception will affect the score on a different belief).

W 2

Recognition of the difficulty of the Join Change Unknown problem and the usefulness ofmanipulatives shows sensitivity to children’s thinking. Insufficient detail is given about the wayschildren would solve the problems for a score of 3. If the response had shown insensitivity tochildren’s thinking, it would be scored 1. (Score 2C)

V 0 Sensitivity to how children might think about these problems not shown.

U 1Child will use visuals but no mention that child will follow action in the problem; thus, only oneaspect of the response is seen as showing sensitivity to children’s thinking. (Score 1B)

T 1The mention of the importance of visuals was interpreted as a minor indication of sensitivity. Nosensitivity to children’s thinking was identified in the remainder of the response. (Score 1B)

Scoren % n %

0 42 26% 17 11%1 74 47% 51 32%2 34 21% 64 40%3 9 6% 27 17%Total 159 159

Pre Post


B6–S8


Belief 6

The ways children think about mathematics are generally different from the ways adults would expect them to thinkabout mathematics. For example, real-world contexts support children’s initial thinking whereas symbols do not.


This item is designed to assess the belief that children think about mathematics in ways different from those adultsmight expect: Respondents are asked whether a word problem concerning fractional parts (without using fractionlanguage or symbols) or a fraction comparison (using only symbols) is easier. Respondents select which of twoproblems would be easier for children to solve, the fraction comparison problem “Which is larger, 1/5 or 1/8?” or acomparison problem set in a real-world context: “Who gets more of a candy bar, those sharing the bar among 5 orthose sharing the bar among 8?” They are also asked to explain their responses. The extensive mathematics-education-research knowledge base indicates that, particularly initially, real-world contexts support children’sthinking whereas symbols do not. (Note that the real-world contexts should be relevant to the lives of the childrensolving the problems.) However, many adults remember having had unsuccessful or unpleasant experiencessolving unrealistic or uninteresting word problems and so tend to believe that solving symbolic problems is easierfor children than solving problems situated in real-world contexts. Respondents who receive the lowest score onthis item state that the symbols are easier for children to understand and indicate a lack of appreciation for the real-world context. In contrast, respondents who receive the highest score not only state appreciation for the real-worldcontext but also recognize that the symbols can be confusing for children. In the latter case, respondents oftenwrite that children might think that one eighth is bigger than one fifth because 8 is bigger than 5 (a misconceptioncommonly held by children).

Note that although in Item 8.1, respondents are asked to rank four fraction problems, for purposes of assessingrespondents’ holding of Belief 6, scorers read explanations of the rankings of only the two problems describedabove (8.1c and 8.1d).


a) Understand


b) Understand



are they same size?


B6–S8

Rubric Scores

0. Responses scored 0 tend to indicate that children more easily make sense of symbols than of real-world mathematicalsituations. This view is reflected in the response that solving word problems is challenging for children.

1. Responses scored 1 indicate that real-world contexts can support children's mathematical thinking, but withoutmention of the potential confusion that children may have when trying to make sense of symbols, in particular, fractionnotation.

2. Responses scored 2 indicate both that real-world contexts can support children's mathematical thinking AND thatmaking sense of symbols, particularly fraction notation, is potentially confusing for children.

B6–S8

Scoring Summary


0

A. Item c is easier (or Items c and d are equally difficult) AND no or little appreciation indicatedfor the use of real-world context to support children's understanding OR

B. Item d is easier BUT explanations are either inconsistent (indicating that c might be easier)or are clearly focused in 8.2 on the teacher’s role in showing students how to solve theproblem

1A. Item c is easier BUT great appreciation for real-world context is shown OR

B. Item d is easier (or Items c and d are equally difficult) AND some appreciation for real-worldcontext is shown or indication given that Item d is easier to visualize

2Item d is easier (or Items c and d are equally difficult) AND Item d is easier because of the real-world context or is easier to visualize ANDItem c is more difficult because numerals are more abstract or confusing (or Item c is moredifficult to visualize).

Comments on Scoring

• When coding, read the response to Item 8.2 first, then read the responses to Items 8.1c and 8.1d.• When the responses to Item 8.1c or 8.1d appear to conflict with the response to Item 8.2, give MORE weight to the

response to Item 8.2.• Although the respondents’ rankings of all items are included with their explanations, the rankings are not used

explicitly in determining a belief score on this item. Use the rankings cautiously if you seek additional informationabout the respondents’ explanations, because some respondents misinterpret the directions and rank the mostdifficult problem as 1. This error is sometimes, but not always, evident from the explanations.

B6–S8

• Examples

1 (Scored re Rubric Detail 0A)q8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_

choiceq8.2 Score Comment

1 easy, just comparing 4 word problems tend tobe perceived as moredifficult

c iseasier

because they are justcomparing.

0 Item c is easier; appreciationfor the use of real-worldcontext to support children'sunderstanding not expressed.

2 (Scored re Rubric Detail 0B)q8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_


1 Easier for children tosee that 5 is bigger than8.

3 It is almost the sameconcept as c.

d iseasier

I would demonstrate achocolate bar and cutit by 5 or 8 so that theycan see the difference.

0 Focus is on the teacher’sshowing the child how to solvethe problem. (Note that thisresponse is scored 0 eventhough the respondentindicated that Item d would beeasier for children to solve.)

3 (Scored re Rubric Detail 1A)q8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_


2 Throwing in the wordfraction makes it moredifficult.

1 Personal experienceswould allow the childrento relate to this problemeasiest.

d iseasier

Children can relate togoing to birthdayparties and of coursethey can relate to howmuch candy they gotcompared to others.

1 Item d is easier; appreciationshown for the real-worldcontext. Response not scored2 because it does not indicatethe difficulties children mayhave understanding fractionsymbols.

B6–S8

4 (Scored re Rubric Detail 1B)q8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_


1 I ranked this problemeasiest because all thechild needs to do hereis draw a pie with fiveslices and eight slicesand then fill in one oneach and they will seethat 1/5 takes up morepie space than 1/8. It isall visual.

2 Again this problem is allvisual if they canvisualize the chocolatebar at each party thenthey will conclude thatJake got more candythan he/she didbecause he had toshare with less people.It is all visual.

c iseasier

I chose c to be easierthan d because in dthey have to read outthe whole problem andsome children have adifficult time readingand understanding ata young age so seeingthe numbers might bea little bit easier forsome.

1 Item c is easier than Item d;but appreciation shown for thereal-world context of Item d.

5q8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_


2 This is a little morecomplicated becauseinstead of anexplanation or wordproblem all you have isintimidating lookingfractions. You have tobe able to picture inyour mind that there is apiece of a candy barthat five people share,and one that eightpeople share

1 This problem lets thestudent visualise what isgoing on with somethingthat they couldunderstand. Theywould be able to picturewhich amount is larger.

d iseasier

You can visualise theproblem in an easy torelate to situation in d.But in c, you have todo all the visualisingon your own.

2 Both some ways real-worldcontexts can supportchildren's thinking andpotential difficulty ofinterpreting fractions arenoted.

B6–S8


Aq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

3 I think this is easierbecause you can showthis problem visually.

3 This problem isconfusing because youwould have to rearrangethe wording in order tocomprehend it. Youalso have to read it afew times and possiblyeven draw the candybars write Jake with thecandy bar divided intofive and friend with thecandy bar divided into 8

c is easier I say C is easier then D becausein C you can see the numbersvisually and in D you have tofigure the numbers out and writethem down adn go through moresteps in order to answer theproblem.

Bq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

4 Fraction have the effectof scaring people. Thekids would think that 1/8would be biggerbecause 8 is biggerthan 5.

2 This would be easybecause all they have tosee is that the lesspeople there are thebigger piece of candyyou'll get

d is easier Both problems are practicallyasking the same thing exceptthere is a story that accompaniesthe fraction problem in letter d.This helps kids visualize theproblem

Cq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

2 I think this problem ispretty simple once thechild has it rxplained tohim/her. They coulduse visual aids or anyother method of viewingwich fractions are largerand smaller.

1 This story problempaints the picture and ismore understandablebecause you know whythe answer is what it is.

d is easier It illustrates the answer so thatyou can visualize the candy barand che amount of children atthe party which helps youvisualize how much candy eachchild would recieve.

B6–S8

Dq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

2 this can be thought ofhow many people youhave to share yourcandy with so it mightnot be that dificult.

1 it is easier to see thatwhen you share withmore people you getless of a piece ofwhatever you aresharing

d is easier d puts the problem into real lifeexperiences where it is easier tosee that you are going to get asmaller piece if you share withmore people

Eq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

3 Because they arecomparing and it isharder than just addingor multiplying

3 The comparison andword problems I thinkare the most difficult fora child

D is easier A child might get confused withthe word problem, though theend result involves the sameproblem as c.

Fq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

2 This concept issometimes hard andmisleading because youwant to think that thebigger number would bethe bigger fraction, but itis the opposite withfractions. This wouldnot be a complicatedproblem as long as thechild knows,understands, andremembers thatconcept.

4 For the most part, wordproblems are moredifficult for children (andadults for that matter) tounderstand, thenumbers are not laid outfor you so you not onlyhave to come up withthe fractions but thenyou have to decide whogot more candy bar.

C is easier I said that c is easier than d butnow that I re-read the problems Iam not really sure that is what Ithink now. Normally I think thatword problems are more difficultfor children but in this situation Ithink that maybe the wordproblem would be easier tounderstand because it is in astory that may be easier tounderstand. When it is in a storylike this they can visually draw acandy bar and divide it up, butwhen it is in just fractions itwould probably be more difficultfor them to draw a picture andfigure it out.

B6–S8


Gq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

4 the concept that 1/8(because 8 is largerthan 5) is smaller than1/5 is difficult forchildren to understand.This totally throws themfor a loop.

1 This is logic andchildren usuallyunderstand this whenyou explain it to them...especially with foodand sharing.

d is easier Well, the concept that a fractionwith a smaller number can belarger is baffoling. 8 has alwaysbeen larger than 5 for childrenand fractions screw it all up.However, when it is in thecontext of "d," it's much easier forchildren to understand.

Hq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

1 This only requires thatstudents understandwhat a fraction is, andcan be shown on apizza or pie.

2 By having this as aword problem, somestudents will be betterable to understand it,while others might getconfused at whatexactly is being asked,as it is a rather lengthyproblem. This requirescomparing fractionsizes, as in c) above,but takes a bit morethinking

c is easier c) only requires that students seethe size difference, somethingthat would be easier than d) ifusing manipulatives, althoughwithout a visual cue, it might bemore difficult than d) since manystudents might confuse the 1/8as being larger since it has an 8.The word problem explains thefractions in a more real worldmanner but it's somewhat difficultto keep the whole story straightin one's head.

Iq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

2 can use fractionmanipulatives torepresent this

1 something concrete torefer to

d is easier concrete references --the childcan relate to the problem. It'snot just numerals on the page.

B6–S8

Jq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

3 this problem involvesgetting a commondenominator, thengetting the correctnumerator and thenalso comparing the twosides

4 this problem mightinvolve drawing out achocolate bar. its along story and involvesdivision and thencomparing twoanswers

c is easier c involves just getting a commondenominator and then multiplyingthe numerator and comparing thetwo answers. d involves readinga long story and dividing andsetting up fractions and thentrying to compare both of them

Kq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

2 Fractions are hard tofigure out which arelarger or smallerbecause they can gettricky. The child has tofind a commondenominator and thenfigure out what issmaller or bigger it getsto be hard.

1 In general, wordproblems are hard tosolve because somuch is written andchildren sometimesgive up before theyeven start. Wordproblems areoverwhelming.

d is easier The child can break down theproblem, one step at a time andthen solve it.

B6–S8


Zq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

3 It is easy to assumethat because 8 isbigger than 5 that 1/8is bigger than 1/5

2 By taking the same problemand putting into a story itmight be easier to understand

D is easier D is a story that takes a conceptand puts it into words that a childcan understand. There may still besome difficulty though.

Yq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

2 THIS WOULD BESECOND BECAUSE ITIS SIMILAR TO D BUTHAS NOT VISUAL

1 THIS WOULD BE THE EASIESTBECAUSE IT GIVES THECHILDREN A VISUAL OF THEFRACTION AND RELATES ITTO REAL LIFE.

d is easier FOR D THERE IS A VISUAL AND AREAL LIFE SITUATION THEY COULDTHINK OF TO SOLVE THE PROBLEM

Xq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

3 need to find acommon denominatorand the compare butsometimes can bedifficult to see whatwould be the commonbetween numbersthat are not obviouslycommon.

1 one candy shared with 5people or 8 people is easierfor a child to see more peopleto share with equals lesscandy for them.

d is easier answer is in the question. Youshare with 4 friends or share with 7friends and which one will producemore candy for you. Most kidsunderstand if they have to sharewoth more friends they will get less.

B6–S8

Wq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

1 WITHMANIPULATIVES IT ISEASIER TOUNDERSTAND WHAT1/5 IS AND WHAT 1/8IS AND THEN TO SAYWHICH IS LARGER.

2 THIS WORD PROBLEM SEEMSEASIER TO COMPREHENDBUT IS VERY WORDY ANDMIGHT LOSE A CHILD'SATTENTION AROUND THE 2NDSENTENCE, WHICH WOULDMAKE THEM UNABLE TOANSWER THE QUESTION. IFTHEY WERE ABLE TO FOLLOWTHE WHOLE ROBLEM, THISWOULD MAKE VERY GOODSENSE TO THEM SINCE THEYUNDERSTAND A CANDY BARSPLIT BY SO MANY PEOPLE,BETTER THAN THEYUNDERSTAND THE CONCEPTOF A FRACTION ON A PAPER.

c is easier THE WORD PROBLEM IS RATHERLONG AND MIGHT LOSE A CHILD'SATTENTION AROUND THE 2NDSENTENCE. OTHERWISE IT ISMORE UNDERSTANDABLE SINCE ITDISCUSSES CANDY BARS ANDNUMBER OF PEOPLE, RATHERTHAN SIMPLY A FRACTION ONPAPER. THE FRACTIONCOMPARISON, HOWEVER, IS MOREEASILY SHOWN WITH PIE SLICESAND WOULD KEEP ONE'SATTENTION MORE EASILY THANSUCH A LONG WORD PROBLEM.

Vq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

2 If children rememberthat the smaller thedenominator is in aproblem like this thelarger it is then theproblem is easy

1 Even though the problemsolving is usually difficult Ithink a child could see thevalue difference in thisproblem easily

d is easier I think c could get tricky if childrenforget that in a fraction the smallerthe denominator is the larger theamount is.

Uq8.1c q8.1c_explain q8.1d q8.1d_explain q8.2_choice q8.2 Score Comment

3 I think this problemis a little hardbecause a childwon't have thelogical thinking tothink about fifths oreights.

3 i think this question alsoneeds the child to have a littlelogical thinking to picture thescene that if there's morepeople then that person isn'tgoing to get more of thechocolate bar because thereare more people to share itwith.

Items c andd areequallydifficult.

Both questions need a person to thinkabout the question and the numbers inthe question. i have to imagine taking1 out of 5 or 1 out of 8, and then inquestion d i need to imagine 5 peopleand 8 people. Ofcourse it doesn't taketime for me to think about the answerbut a child would probably need timeto think about the answer.

B6–S8



A 0 Item c is easier; appreciation not expressed for real-world context

B 2Item d is easier, appreciation shown for the real-world context; potential confusion of the symbol1/8 with 8 and 1/5 with 5 recognized

C 1Item d is easier; appreciation shown for the real-world context, but difficulties of interpreting themeaning of the fraction symbols not mentioned

D 1Item d is easier, and appreciation shown for the real-world context; no mention of difficulties ofinterpreting the meaning of the fraction symbols

E 0 Item d is easier; no indication of appreciation for real-world contexts.

F 1Item c is easier, but appreciation indicated for this particular word problem in 8.2. (Refer toscoring note that if responses to 8.1c or 8.1d conflict with the response in 8.2, more weightshould be given to the response in 8.2.)

G 2Item d is easier; appreciation stated for the real-world context (food and sharing) andrecognition shown for the potential confusion children might have understanding the meaning ofthe symbols 1/8 and 1/5 in relationship to the symbols 8 and 5.

B6–S8


H 1Item c is easier; appreciation shown for real-world context and for the potential confusion of thesymbols. This respondent also worries about the length of the problem, but because of supportfor real-world context to help children understand, response is scored 1.

I 1Item d is easier; the "concreteness" of the word problem to support children's understanding isnoted. Although the sentence "it's not just numerals on the page" mentions symbols, it does notindicate that the symbols might be confusing for children.

J 0Item c is easier; appreciation not shown for the real-world context to support children'sunderstanding

K 0Item d is easier but appreciation for real-world context unclear. Response 8.2 is vague, andResponse 8.1d states that word problems are too challenging for children.

Z 2Item d is easier; (weak) appreciation shown for real-world context (in 8.1d and 8.2) AND for thedifficulty students may have interpreting the fractions (in 8.1c). (Weak 2 because theappreciation for real-world context is qualified.)

Y 1

Item d is easier; the statement that real-world context provides a visual is interpreted to meanthat the context helps the child to solve the problem. (This response is not scored 2 becausealthough the respondent writes that c is similar to d except that c has no visual, we did notinterpret this contrast to show that the respondent thought that the lack of a visual couldpotentially make the problem confusing for children to understand. The sentence, instead,seemed to convey how items c and d are different.)

X 1Item d is easier; appreciation shown for the real-world context but not for the difficulties inherentin the symbols themselves (but instead for aspects of the procedure of finding a commondenominator).

B6–S8


W 1Item c is easier; in general, appreciation is shown for the real-world context, although therespondent states that this particular context is too long. This response is not scored 2,because Item c is selected as easier than Item d.

V 0

Item d is easier, but Response 8.2 concerns the difficulty of fraction comparison, not the waycontext supports a child’s thinking. (Although context is weakly supported in 8.1d, refer toscoring note that if responses to 8.1c or 8.1d conflict with the response in 8.2, more weightshould be given to the response in 8.2.)

U 0Items c and d are equally difficult; appreciation not shown for real world context’s support ofchild's understanding. The respondent mentions the difficulty thinking of fifths or eighths, butbecause of the lack of support for the real-world context, the response is scored 0.

Scoren % n %

0 96 60% 75 47%1 40 25% 51 32%2 23 14% 33 21%Total 159 159

Pre Post


B6–S9


Belief 6

The ways children think about mathematics are generally different from the ways adults would expect them tothink about mathematics. For example, real-world contexts support children’s initial thinking whereas symbolsdo not.


In scoring responses for this segment, consider whether respondents suggest the use of representations otherthan symbols to teach division of fractions. Responses indicating that work with manipulatives or visuals shouldprecede work with symbols are interpreted as providing strong evidence of this belief. Respondents whosuggest that the teacher provide an explanation, more practice, or both are considered to provide no evidence ofthis belief. We interpret suggestions that symbols and other representations be taught simultaneously as someevidence of this belief.

Focus attention on the responses to Item 9.6 and then consider whether in 9.1 the respondent discusses theimportance of using representations in teaching fractions. Other responses should be skimmed for any relevantinformation. Reliability on this rubric was quite high because the rubric is relatively objective.





Yes No









http://coe.sdsu.edu/survey/videos_wmv/IIIb.wmv

http://coe.sdsu.edu/survey/videos_wmv/IIIb_56K.wmv



B6–S9

Rubric Scores

0. Responses scored 0 indicate that for students to master this procedure requires only work with symbols.Although some respondents may express concern about conceptual understanding, they do not mention that otherrepresentations will help children better understand the concept. They are not considered to show evidence ofholding this belief if they do not suggest that the teacher present the concept using some other means.

1. Responses scored 1 indicate that the teacher should provide the children with other representations in addition tothe symbols.

2. Responses scored 2 indicate that children’s reasoning for division of fractions will be better supported with otherrepresentations instead of or prior to manipulation of symbols.

B6–S9

Scoring Summary


0• Practice more (with no mention of teacher intervention of any kind)• Practice more of the same kind with some support from the teacher.• The teacher should explain why the procedure works along with providing more practice.

1 • The teacher should provide a context or visual aids in conjunction with the algorithm.Difficulties children will have with symbols not acknowledged.

2• Children can reason through such problems, and the teacher should support this reasoning

with the use of visual aids; the algorithm should be introduced after the use of visual aids, ifat all, or children will have difficulties with the symbols.

B6–S9

Examples

1q9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

The childseemed tounderstandwhat she hadbeen taught

That is you flipthe other andmultiply youcan get thecorrect answer

Yes, I think shewould be able tobecause she didnot show difficultyor hesitation indoing the problemby herself.

The child wasunsure of how tosolve the problem.She lackedconfidence

Some kids knowhow but just donot believe theydo and need tobasically havetheir mindsrefreshed andencouraged tobelieve they can.

Give them a reminder torefresh their minds

0 No suggestion that theteacher userepresentations thatmay be moremeaningful to children.


I was verysurprised tosee that afterseeing threeexamples andonly doing oneproblem on herown, sheremembered allthe steps tosolving theproblem.

I think the childonlyunderstandswhat theteacher hasshown her todo. Sheprobablydoesn'tunderstandwhy.

Yes, I think thechild would beable to solve asimilar problemin a few days,as long as shehad some typeof homeworkinvolving thesetypes ofproblems orsome type ofreview.

The child could notremember the stepsto figuring out theanswer to theproblem. She wasobviously onlyshown the few daysbefore with noexplanation of whyand not given anymore practice atthis type ofproblem.

I would say about5 because notmany children willremember thosesteps to solvingthe problemwithoutunderstandingwhy they had todo those certainsteps.

If the teacher asked thechild more questionsand had given the childexplanation as to whythose steps took place,I think more childrenwould remember more.

0 No suggestion that theteacher userepresentations thatmay be moremeaningful to children.


She rememberedthe steps insolving theequation, andrememberedwhat the teacherhad told her.She was able tolearn them andapply them toother problems.

I don't think shequite understandswhy you flip it,but that itssomething you doto solve theproblem. Sheunderstands thatyou flip thefraction and thenmultiply it straightacross.

Yes. I haveconfidence thatthis child couldsolve similarproblems becauseshe knows thesteps andmethods of howto go aboutsolving such aproblem

She forgot thesteps of how tosolve theproblem.

On furtherthought, if theprocess isn'tpracticed often, itcan be forgotten,especially sinceits just learned.

Explain why the problemworks the way it does.Possibly create a senarioto explain what is goingon, or to use more visualaids. Also, assignhomework on the subjectso the students cancontinue learning theprocess and memorize itat home, and then teachit more in class.

1 Respondentacknowledges thatmanipulatives may helpsupport children’sthinking but does notacknowledgedifficulties children mayhave with symbols.

B6–S9


one on oneinstruction is nice

- no, she ismimicking themath theinstructor justshowed her

Yes. if thedenominatorwas the same,no if they aredifferent,because shewas copyingthe way theinstructorshowed her asa format notas anunderstanding

- she was not ableto complete theproblem becauseshe never fullyunderstood theconcept she wasonly repeating whatthe instructorshowed her at thattime.

have tangibleconcepts to touchand visualize likepie charts in colorto show 1/3 and1/5

1 Respondentacknowledges thatmanipulatives may helpsupport children’sthinking but does notacknowledgedifficulties children mayhave with symbols.


confusing to child,boring,meaningless

nothing No. She wasvery good atmemorizingthe algorithmin the shortperiod oftime, but itwill not staywith her.

painful to watch.Child had nounderstanding ofwhat was beingasked. Child took along time to writethe 6 because sheknew from thebeginning that shedidn't know how todo the problem.

no understandingof what thedivision reallymeant

I don't know how youshow division withmanipulatives (yet) butyou would have to startthat way. The childrenwould have to havemany experiences tounderstand theconcept before evenshowing them thealgorithm.

2 Respondent indirectlyindicates thatalgorithms are difficultfor children to makesense of andadvocates use ofmanipulatives to teachthe content.

B6–S9


The childunderstood howto use thealgorithm tosolve theproblem, but shealmost seemedlike a robot. Sheused the exactsame words andsteps as herteacher. Shedidn't need tothink at all, and Idon't think sheunderstood whatshe was doing.

Not much. Ithink she wasjust repeatingwhat herteacher hadshown her,without anyunderstandingof division offractions or whyshe was doingwhat she wasdoing.

No. She mayremember somesteps of theformula, butuntil she clearlyunderstandswhy she isdoing what sheis doing, shewon't be able tosolve similarproblems,especially ifthey are just alittle bitdifferent thanthe ones thatshe had beentaught to solvewith thealgorithm. Thatalways throws achild off track,when theproblem theyneed to solve isdifferent thanthe ones theylearned how tosolve with thealgorithm,unless theyclearlyunderstandmathematicallywhy they aredoing what theyare doing.

That was what Ithought was goingto happen. Thechild had been ableto repeat what theteacher had donethe day before,without anyunderstanding, likea robot. But whenshe had to comeback the next day,she didn't knowwhat to do, or whyto do it. It had justbeen numbers and aformula to her, notunderstanding.

I think that somechildren wouldremember what todo, just becausethey may have agood memory.Others may justtake a goodguess and beable to solve it.

I think the teacherwould need to usesome visual aids anddrawings to get thechildren to betterunderstand it. Evenwhen I was a child,before I took my Math210 class, I used tomake drawings to try tofigure out how to solvemath problems. Thatwas what worked bestfor me, and wouldprobably work best forat least some otherchildren. She coulduse real life examples,but not word problemswritten down on paperyet, and have themsolve those real lifeproblems. Then theteacher needs to relatethe real life problems tothe ones on the paper.If she can clearly getacross the message asto why the children aredoing what they'redoing, I think that theywould understand andbe able to solve similarproblems in the future.

2 The respondentsuggests using real-lifecontexts to supportchildren’s thinkingbefore attending to“word problems writtendown on paper.”

B6–S9


The teacher waspleasant andhelpful, but thiswas simply rotemanipulation ofabstract symbolsthat had little ifany meaning tothe child.

Based on whatwe say here,none at all(although shemight knowmore -- shenever had thechance todemonstrateanyunderstanding).

Yes. The lengthof time here issignificant andwhat else thechild does in themean time issignificant. If thiswere 3 monthsafter this sessionand the child haddone no otherfraction lessons,the odds of recallare minimal. Thisis a pretty easyprocedure

It is clear the childforgot thealgorithm and hadno idea what to do.

As I noted before,there are a lot ofmitigating factors.

I would take a singleobject (fraction bar,candy bar, fractioncircle) and work on theidea of how many 1/3'sthere are the singleobject. I would then goto 6 of those objectsand how many 1/3'sthere are in the 6objects.

2 Comments on how itcould be developedconceptually in Item9.6


I thought the teacherdid a fine job ofexplaining how to dothe problem andshowing the studentthree practice problemsthen having her do oneon her own andexplaining how she didit was a good way

she knowshow to findthe answerbut she mightnot knowwhat iymeans orlooks like

Yes becauseshe was ablesto do one onher own andexplain back tothe teacher howshe did it

I thought itwas sad thatshe didn'tknow whatshe wasdoing and itwas onlythree daysafterwards.

I think only 25because if this girlknew how to do it andwas able to tell theteacher how she did it, then forgot threedays later, I just thinkabout the kids thathad a hard time doingot the first time

I would hope theteacher would practicewith the studentseveryday until they allwere confident aboutdoing the problem

0 Directly suggests tohave teacher providepractice.


The child didn'tseem tounderstand whatshe was doing.She was morefocused on herlines being straightthen doing theproblem.

Not much. I thinkthat this was herfirst experiencewith division offractions, and shewould need morepractice tounderstand theconcept.

No. She couldbarely rememberhow to do theproblem the sameday. I don't thinkshe couldremember thesteps in a fewdays.

The studentforgot whatto do.

I don't think thatmost children couldremember a newmethod after such ashort exposureperiod.

Repetition. I thinkthe repetition willhelp students overtime. Also,manipulatives arehelpful.

1 Although respondentstates that repetitionis key, adds thatmanipulatives couldhelp.

B6–S9


The child was ableto pick up themethod of dividingfractions after onea few trials with theinstructor.

I do not think thechild reallyunderstands awhole lot aboutdivision offractions. shehas only had thisone lesson andwithout furtherlessons she willforget everythingshe has beentaught today.

N0. She onlywent through acouple problemswith theinstructor.Children need todo the samekinds ofproblems overand over so itsinks in.

The child couldonly vaguelyremember whatshe wassupposed to do.If she knew moreabout divisionproblems andunderstood themmight have abetter shot atsolving thisproblem.

This child is verytypical. With my fieldstudy of kids anddivision problems, iknow you have to drillit into them beforethey can go home onthe weekend, comeback and still be ableto solve it.

Tell them whythey have to flipone of thefractions andwhat is going onwhile they areteaching it.

0 Suggests that teachertell student why theyhave to "flip."


the teacher had adifferent approachwhen teaching thechild this problem.she used differentcolored makers,and did each stepslowly in explainingwhy she differentsteps. this helpedthe student tounderstand

the process ofsolving themand that theyare similar tomultiplicationproblems

yes, with practiceand constantreinforcement, shehas it down pat

the child was in abind as to how tosolve the divisionproblem

not very many constantreinforcement,use ofmanipulatives,

1 Manipulatives toreinforceprocedure––at sametime.


With the teachershowing the studentthe steps to thistype of equation andgoing over it withher several timesbefore letting her doit on her own helpedher do andunderstand theproblem on her own.

The studentsunderstandsthat the fractionis a wholenumber andtheyunderstand theprocedures onhow to get thatwhole numberout of a fraction

Yes. I would thinkso but the studentmay have difficultybut they wouldremember most ofit.

The student did notremember what todo.

I see that theywould have troubleand the best thingto do is go over itwith again.

I would say givethem problemslike thateveryday and goover it with themstep by step.let it be areoccuringproblem so theycan get familiarwith it.

0 Suggests giving lotsof practice.

B6–S9


at first the problemseemed to be hard,i was surprisedthat she learnedvery quickly

the multiplicationprocess, youneed tounderstandmultiplicationfirst then startwith division

Yes. becausethe teachershowing manyexamples andwhen she was,she had the childrepeat and followwhat she wasdoing, so thechild learned fast

i felt really bad forher when she couldnot answer it, i feltbad that she forgot it

i believe tenbecause although itis a low answer itwould happenbecause there wasno practice and it issomething that achild might not beable to remember atthat time

keep practicingthe problems,give homeworkassignment andmaybe do classwork

0 Practice.


I noticed thatthe child waslearning how todivide fractionstoday. I alsonoticed that thechild learnfractions bylearning thealgorithm andnot through theuse ofmanipulatives orany other sortof guidancematerials tohelp the childunderstand why

I think the childunderstands verylittle aboutdividing fractions.I'm certain shedoes notunderstand whyshe has to "flip"the fraction norwhy the operationchanges fromdividing tomultiplying. I alsofeel that when theinterviewer askedwhether or n

No. She clearlywould notremember thealgorithm becauseit's too vague anddoesn't allow thechild tounderstand thesteps. I'm sureshe would get thesteps mixed-upand forget someas well.

She does not knowhow to solve theproblem because shewas not able to relateto the problem in acontext she couldunderstand andvisualize. I think thechild gained nothingfrom the previouslesson with theinterviewer when theinterviewer tried toexplain.

This child I ampretty certain isvery typical.

The teacher shouldnot introduce thealgorithm soquickly into thelearning processfor the kids. Ithink the teachershould work slowly,first by usingmanipulatives orother such tangibleobjects where thechildren could "see"for themselves howthe pr

2 Suggest notimmediatelyintroducingalgorithm; suggestsprogression.Suggests use ofconcrete examplesto illustrate/explain(manipulatives,diagrams).

B6–S9


Aq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

The child wasgetting noconceptualinstruction.

She onlyunderstands atechnique to getan answer.

No.with noconceptualgrounding orframework, thistechnique wouldbe easilyforgotten.

It's what Iexpected

Maybe a fewchildrenwouldremember.

The teacher wouldneed to explain anddemonstrate theconcept of divisionof fractions with real-life examples.

Bq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

there should havebeen way morebuilding up to thislesson. the girlknows how to gothrough the motionsbut i don't think shereally understandswhat she is doing.she doesn't evenhave any idea ofwhat dividingfractions really is.or what she is doing

absolutelynothing, butshe can solvevery basicones. that isgreat that sheknows analgortithm but ina week fromnow i wonder ifshe willremember it.

No.if youdon'tunderstand why arewhat youare reallydoing, itmakes itdifficult torememberbecauseyou neverreallyknew whatyou weredoing inthe firstplace.

interesting. ibet if therehad beenmore of awhy lessonand shereallyunderstood ita couple ofdays ago,she wouldhaveremembered.instead shenever reallyunderstood itand that isapparent inher nothaving anyidea on howto solve it afew dayslater

i am notsure howmany wouldremeber, buti would thinkit would be avery smallnumverwouldremember

have more of aconceptual lessonbefore thealgorithm. i think itis ok to showstudents shortcutsafter you know thatthey reallyunderstand whatthey are doing. ifthere had maybebeen drawings,pattern blocks, orsome type ofmanipulative to helpthe themconceptually graspthe concept.

B6–S9

C

Dq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

It was sadbecause I knowthat mostchildren receivethat same typeof meaninglessinstruction. Itturns children offto math. Itmakes mathseem like somemystery, somemagical thingyou do. It doesnot give a childanyunderstanding atall.

She probablyhas no realunderstanding.

No. Theteacher gavethe child analgorithm tofollow. Thechild doesn'tknow why shechanges thedivision tomultiplication orwhy she "flipsthe number."She was able tomimic the stepsfor thisinterview, but Idoubt she wouldretain thealgorithm steps3 days later.

It didn't surpriseme at all. Thechild had nounderstanding ofdivision offractions only analgorithm to follow.Until the algorithmis drummed intoher head by doingit (perhaps evenmindlessly) manytimes, she wouldbe unable to do thealgorithm. Oh, wewaste so muchtime in theclassroom teachingalgorithms beforeany mathematicalunderstanding hastaken place.

Somechildrenhaveexcellentrecall evenfor amindlessalgorithm !

The childrencould do reallife divisionproblemsthat madesense to thechildren andforgo thealgorithmaltogether.

q9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score CommentThe teacher did agood job ofexplaining herselfand the child wasvery attentive.

I think sheunderstands todivide you haveto reverse thesecond fraction,but I don't thinkshe understandswhy she has to.

Yes. Shehad a goodunderstandingand if shetried again ina few days itmay take hera little time,but I thinkshe'll get it.

She forgothow to do theproblem.

The child isverytypical, andwithout alot ofpracticethey will allhaveproblems.

Using anumber ofdifferentfractions insimilarproblems.

B6–S9

Eq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

I was amazedthat the girlwas able to getthe problemcorrect on herown after onlya fewexamples. andshe was alsoable to explainwhat she wasdoing.

I think theremight be thepossibility thatthe girl wasable to copywhat theteacher didbecause it wasright in front ofher and freshin her mind.

No. Maybe notsolve it to getthe rightanswer rightaway. it mighttake some timeand a littlerefreshing onhow to do it.

The girlcouldn'tremember whatto there wasn'tany kind of anexample for herto look at andcopy so it washard.

Children aregood atcopying adultsto give themthe answer thatthey want. butwhen theyhave to do iton their ownwithout anytype ofexample orhelp it is hard.

The teachershould doreview beforegiving theproblem to thechild to do ontheir own.Maybe doingthe problemtogether,having thechild tell theteacher what todo.

Fq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

The never showedthe girl with drawingswhat they weredoing. The couldhave drawn 4squares, then dividedthem into thirds andcounted up thenumber of thirdsthere were.

I think that the childunderstands how todivide them usingnumbers, but shehas no idea why sheis doing it. She hasno means of applyingthe data to everydaylive.

Yes. Thevideo saidthat sherehearsed afew moreproblems.Hopefullyshe willhavememorizedthe process.

She hadforgotten howto divide withfractions. Ialso don'trememberseeing a timeframe, was thisthe next day?A week later?A month?

Her matchskills arenot out ofthe norm.

I think thatshowingillustrationsof what isbeing doneor providingmanipulatives wouldhavehelped.

B6–S9


Gq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

The child mayhave been a littleaprehensive aboutdiving fractions butwatching the stepsand hearing theexplanation alongwith working somewith the teacherand some withoutthe help allowedher to learn thesteps.

She may notunderstand all thewhys, like why youbut the four over aone or why you flipthe second fractionbut she know thebasic steps tosolving that type ofproblem. She maynot be able to solvea more difficult onethough.

Yes. Shemay need alittlerefreshing tocomplete theproblem and itwould help ifshe had hadpractice inthe daysbetween but Ithink theprocesswould comeback to her.

The childwasconfusedabout howto go aboutsolving theproblem.She wasunsure ofthe stepssheneeded totake.

I think thisprobably verytypical. Thechildrenrememberlearning toprocess butaren't sureabout theirability tocomplete theproblem.

The teacher couldprovide moreexplanation aboutwhat is involved individing fractionsand what each stepmeans. Also, if theteacher was able torelate it tosomething thechildren could relateto or visualise, itmight be easier forthem to remember.

Hq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

I thought thatshe did verywell figuring outhow to do it onher own. Afterthree problemsshe figured itout withoutassistance. ithought therecould havebeen somemore hands onexperience thefirst time shewas beinghelped andthen i felt thatshe was stillnot veryconfident indoing theproblemherself.

I dont knowif sheunderstandsanything,but how todo it onpaper. ithought thatthe processcould havebeenexplinedbetter,maybe withthe purposeof why thesign shouldbe changedand why the2nd fractionneeded tobe flippedover.

No.Withouttheassistance of theexamplein front ofher, idon'tknow ifshe wouldbe able tofigure outa problemon herown. I amnotconfidentthat shetotallyunderstands theprocess.

The studentwas unabletorememberthe processthat shehas beendoing thefew daysbefore.She nolonger hadtheexample infront of herto follow orthe helpfrom herinstructor.She hadneeded abetterlearningprocedure.

I think that it is veryhard to remeber aprocess you areunfamiliar with. The girlhad been given thechance to learn it, butit was not explained allthe way, and all shehad was a formula. Idon't think manychildren would be ableto remember how to doit. Not only is math notnumber one on theirminds, but I think it isimportant, if you wantthe students toremember something,that you continue towork on it with them,until they are confidentthat they know how todo it on their own.

One of the things that Inoticed right away, was theteacher took over incompleting the firstproblem. i think by writingit out as many times aspossible there is a betterunderstanding. Also shedid not give direction aboutwhy they were changingthat that division sigh tomuliplication to or at whythey had to flip the 2ndfraction. I think it is veryimportant that they have afull concept on what theyare doing. Then when theycoukd see that at the endthe young girl was stillhesitating with her answer,go over more problems untilthey feel comfortable ontheir own.

B6–S9

Iq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

At first I thoughtmaybe theteacher was alittle pushy but Ithink throughoutthe video sheeased up andallowed the childto answerquestions. Iliked how shepraised her fordoing the rightthing andexplaining it

I think sheunderstandshow to dothem. Shemay notunderstandthe wholeconcept ofthem. Shedoes knowhow to followthe porocessor steps tosolving adivision offractionsproblem.

Yes. The teacherallowed her to exlpinback to her whatshe did to makesure she knew whatshe was doing. Alsoif she missedsomething theteacher made sureshe knew what shewas doing by askingquestions.

The childsat andstruggledwith tryingtorememberhow to dotheproblem.She forgotwhat to dobut she didtry to dosomethingwhich isalways agood sign.

I think a goodamount wouldbe able to do itbecause ofbeing visaullearners. Ithink she was atypical childbecause it doestake a while fora child toremember howto do a problem.It takes timeand lots ofpractice.

I think theyshould havecontinued topractice withthe child.She alsoshould ofhad the childtake someof theproblemshome tokeep it freshin her mind.

Jq9.1 q9.2 q9.3 q9.4 q9.5 q9.5 Score Comment

the child was ableto do that problemsince it was similarto the example

not much, onlybe able to do iton paper

No.maybe, imnot sure,but if i wasto guessshe wouldprobablyforgetbecauseshe is justfollowinganalgorithmrule andnot fullyunderstandthenumbersrepresentation

she alreadyforgot thethe rules ofalgorithm;just as iexpected

thats just aguess

visualrepresentationfirst, thenalgorithms

B6–S9

Kq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

The thing thatstood out tome was the thechild seemedto be repeatingwhat shesaw.She hadno realexplination forwhat she wasdoing, and shedidn't even askany questions.

I think thatthe childunderstands nothing.

No. The child wassitting there withan example of howto solve that typeof problem, butshe didn't learnanything and hadno real explinationfor how or why it isdone that way.

I thinkthat sheforgot howto solvetheprobelmbecuaseshe neverreallyunderstood it in thefirst place.

I think that therewould be the fewkids who knewhow to memorize,or actuallyunderstand theproblem, but forthe most part kidswouldn't rememberhow to solve theproblem becausethey didn't have areal understanding

I think that usingmanipulative wouldbe a good method,becaue then thechild couldvisualize theproblem. Alsotelling them whatthey are doing andwhy they are doingit might also help.

Lq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

well at first theteacher didn' t do agood job ofexplaining teproblem maybeafter sometime thestudent was ableto do it on her ownbut the way theteacher aproachedthe problem sheseemed kind ofmean maybe alittlemore enthusiasm

that all youhave to do isflip the secondnumber andmultiply across

No. theway theteacheraproachedit was sodry that itgave noreason forthe child toremember it

the child wasnot able torememberwhat she hadlearned threedays before

children need areason toremember if theproblem wastought with moreenthusiasm andsoemthing thatwould makethem rememberthey would beable to do itthree days later

be moreenthisiastic likeshe want s tobe there andnot be so dryand mean

B6–S9


Zq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

The studentdid rememberthe correctalgorithm but Idon't know ifsheunderstoodwhat she wasdoing or whyshe flippedand thenmultiplied.Where wastheunderstanding.

I think sheonlyunderstands how thealgorithm issupposedto work. Ithink sheonlymemorizedthe stepswithout anymeaningbehindthem,

No. I havelearned thatsimplymemorizing howto do somethingis not a goodway to rememberhow to dosomething. Youmust understandhow somethingworks and why itworks in order touse it later.

She had onlymemorized thealgorithm ofdividing fractionsfor a short time.She had nounderstanding ofwhat to do withthe numbersbecause thealgorithm gaveher no meaningas to what shewas actuallydoing.

I onlythink thatthose fewchildrenwith reallygoodmemorieswouldknow howto do itthree dayslater.

The teacherneeds to givethe studentsmeaning behindthe algorithm.She could showthem withpictures orblocks first andthen show themhow thealgorithm ties inwith it.

Yq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

I think that thechild in the clipwith practice canlearn the method ofdividing andmultiplyingfractions.

Nothing quiteyet, maybeplacing a oneunder a wholenumber .

Yes, if shepracticedthe methodshe learnedfrom herteacher.

She struggledonrememberingwhat to dowith thefraction asfar as what todo with theone and thethree.

I would saythat lessthan 50percentcould solvethis a fewdays later.it would takepracticebecause itis dealingwithfractionsrather thanjust wholenumbers.

Try to explainmore thoroughlyon what to doand why thedivision signturns into amultiplicationsign whenputting a oneunder the wholenumber andallow thechildren topractice more.

B6–S9

Xq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

She seemed tofollow theprocedure verywell. she mighthave beenunsure of whyyou flip theproblem though

that it reallyisn't division ,it seems morelikemultiplication

No. shewillprobablyforget thesteps andnot reallyknow whyyou dividefractionswithmultiplication

the child forgothow to solve theproblem. I thinkshe knew sheneeded to changethe sign andmaybe knew sheshould flip thefraction but justcouldn't reallyremember how orwhy

I think this isvery typical.some kidsmight pick upon it butmost wouldneed morepractice

more visuals,longer lessonand morepractice problemat home and inclass. also theyprobably needmoreexplanation ofthe steps.

Wq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

The child didexactly whatshe was told,no more or noless. sheseemed passiveand didn't askmanyquestions.

I don't know ifshe knows whyshe flips thefraction or whyshe multiplies. Idon't think shereally understandwhat she isactually doingwith the fractions.She was goingthrough the stepslike she watchedthe teacher. ithink that shesolved theproblem becauseof watching theteacher.

No. shewillprobablyforgetthemethodand willneed theteacherto showher howto do itagain.

The childforgot themethodshe wasshown afew daysago. thisshows thatshe didn'tunderstandtherelationshipof the 2fractionsand whatshe wastrying tosolve.

About 20% of thechildren would beable to solve thisproblem laterbecause they weretaught too quicklyon how to performthe problem, nothow to understandthe problem. Therewould be a few thatwould rememberhow to do theproblem becausethey got a betterunderstanding orthey have a bettermemory.

The teacher needsto draw visuals ormake up a story sothe children canrelate to theproblem. If sheexplains it in a waythat they arefamiliar with, thenumbers and thesymbols wont beso foreign to them.they couldremember the storyto guide them insolving problemssimilar to this.

B6–S9

Vq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

I thought thatthe process ofshowing thechild how tosolve theproblem was welldone. Thereasoningbehind thisproblem is notclear and wasnot explained.

I think thatthe childunderstandsthe flip andmultiplyprocess but Ido not thinksheunderstandswhy thistakes placeand what ithas to dowith divisionof fractions.

No. I thinkshe wasmimickingthe originalproblemsshe solvedwith theteacher andin threedays, shewill not havethatguidanceand theprocess willbe lostbecausethere is nomeaningbehind it.

The child wastrying toremember theprocess of howto solve theproblem andsherememberedsomethingabout switchingthe divisionsign forsomething elseand then shecould notremember whathappened next.

The meaningand processbehind thisdivisionproblem wasnotexplained sothere was noreason toremember it.

I would let thestudent try to figureit out on her ownfirst, then I woulduse drawings orsome type ofmanipulative toenhance themeaning behind thereason for solvingthis problem, thenshow the process.Then maybe thechild wouldremember better.

Uq9.1 q9.2 q9.3 q9.4 q9.5 q9.6 Score Comment

She understoodit quite well.

the process No. She onlypracticed oneby herself

She couldntrememberwhat theprocess was.

You need topractice moreof them beforethey will knowit.

Give themmoreproblems.

B6–S9



A 1Use real-life examples. Difficulties children have with symbols not acknowledged; that useof real-life examples should precede the work with symbols not mentioned.

B 2Children should be introduced to the symbols only after they have had opportunities toexplore the concept with drawings or manipulatives.

C 0 Only more practice needed.

D 2Use real-life contexts instead of the algorithm; children have a hard time understandingsymbol manipulation.

E 0 More of the same teaching suggested.

F 1Use of a different approach to teaching the content suggested, but difficulties children havewith symbols not acknowledged directly.

G 1 Use visuals along with an explanation. Difficulties with symbols not discussed.

B6–S9


H 0 No suggestion for teacher to use other representations to teach the concept.

I 0 Continue practice. No mention of using other representations.

J 2Although short, the response indicates that visual representations should come beforework with symbols.

K 1 Manipulative use should accompany work with symbols.

L 0 Only concern mentioned is with affective issues.

Z 2Use blocks first, to support children’s reasoning and then move on to the algorithm. Thisresponse is somewhat “tricky” because, at first (give the meaning behind the algorithm) itseems to indicate that one would use the blocks in conjunction with the algorithm.

Y 0 Explain why the steps are performed; provide practice.

X 1Use of various representations suggested, but no indication that these should be usedbefore emphasizing symbols.

B6–S9


W 1Visualization will support children’s’ thinking, and symbols are difficult for children.Respondent recommends that the visualization be shown in conjunction with the symbols.

V 2 Use manipulatives to support children’s thinking before sharing the algorithm.

U 0 More practice.

Scoren % n %

0 136 86% 105 66%1 19 12% 44 28%2 1 1% 10 6%Total 156 159

Pre Post


B7–S5


Belief 7

During interactions related to the learning of mathematics, the teacher should allow the children to do as much of the thinking as possible.


This is the only segment on the survey that is not situated within a particular mathematical context and is instead moregeneral. This fact may contribute to this rubric’s being one of the more difficult rubrics to code. It requires closescrutiny because the responses to this segment vary greatly We identified eight kinds of responses to this segment.

The segment was designed to measure (a) the respondents’ beliefs about the role of the teacher and (b) thedegree to which respondents state they are willing to share the authority in the class with their students. Item5.0, about the respondents’ personal experiences with problem solving, serves to stimulate their thinking.Responses to Item 5.0 occasionally provide evidence about the belief, but coders should focus on answers toItems 5.1 and 5.2 because they tend to be more informative. In coding this segment, attend to how therespondents plan to use children’s thinking in their teaching. In coding, we asked ourselves, “Does therespondent indicate that the students’ approaches will be as legitimate as the teachers’ approaches?” We takeas strong evidence for this belief indications that children’s thinking will be central to the respondent’sinstruction. We infer a respondent’s plan to provide children with problem-solving opportunities but not use thechildren’s thinking in instruction as weak evidence of this belief. Such a respondent plans to allow students todo mathematical reasoning but will not use it as an integral part of instruction. A respondent who intends to usechildren’s thinking in instruction, but only infrequently, is considered to provide evidence of holding the belief.


Yes

No

B7–S5

Rubric Scores

0. Responses scored 0 indicate that students will never be asked to solve problems without first being shown how todo so. The respondents state that children need to be provided with approaches and cannot devise them on theirown. They do not show evidence of believing that children should think for themselves.

1. Responses scored 1 indicate that children will be asked to solve problems without being shown how to solvethem, but do not indicate that this work will be an integral part of instruction. The respondents will provide childrenwith problem-solving opportunities, but only in limited circumstances: for assessment, to stimulate thinking beforeformal instruction, or after some introduction by the teacher. In some cases they state good reasons for allowingchildren to solve problems for themselves but also state that the teacher should show some strategies as well.

2. Responses scored 2 indicate that children will be asked to solve problems without being shown how to solve themand that the children’s approaches will be used as an integral part of instruction. The respondents, although theynote the importance of this practice, may use it infrequently. Some responses indicate that children’s approacheswill be as legitimate as the teacher’s and will be useful to other children.

3. Responses scored 3 indicate total commitment to students’ solving problems without first being shown how to doso. Problem solving will be an integral part of the respondents’ instruction. The response includes a soundrationale for the importance of children’s devising strategies and solving problems themselves and doing so often,if not always.

B7–S5

Scoring Summary


0 • Don’t let students think on their own.

1

A. Provide students with a little instruction first and then let them think.B. Let students think on their own then tell them “my way.” May include a good reason for having

children solve problems on own.C. Let students think on own so teacher can assess understanding.

Does not exhibit appreciation of importance of children’s devising strategies for themselves(beyond correcting wrong answers).

D. Let children think on their own to motivate them or to develop their self-esteem.Does not exhibit appreciation of importance of children’s devising strategies for themselves(beyond correcting wrong answers).

2

A. Let students think on their own.Exhibits appreciation of importance of children’s devising strategies for themselves.This practice will be used sparingly; OR rationale is not well developed

B. Let students think on their own.Students’ approaches will be as legitimate as the teacher’s.Unclear how much this practice will be used.

3• Let students think on their own.

Exhibits appreciation of reasons children’s devising strategies for themselves is important.This practice will be common in the classroom—used often.

B7–S5

Examples

1q5.0 q5.1 q5.1_explain q5.2 Score Comment

i don't remember ever solvingproblems without the teacherexplaining it (i was in elementaryschool 25 years ago!)

no it just gets them frustrated and mathis hard enough!

not asked 0 Respondent will not give childrenopportunities to devise solutions forthemselves.

2 (Scored re Rubric Detail 1A)q5.0 q5.1 q5.1_explain q5.2 Score Comment

It was confusing because I had tothnk of a way to solve it even thoughI didn't realy know what I was doing

No Because the child needs a model firstso thtat they get ad idea of what theyare doing and then they can think forthemselves and make their owndecisions

Not asked 1 Children can do reasoning on their ownbut only after they have had someinstruction from the teacher.

3 (Scored re Rubric Detail 1B)q5.0 q5.1 q5.1_explain q5.2 Score Comment

i liked ot sometimes. when i wasyoung i loves math and puzzles andtrying to figure stuff out on my own. ihad a teache who would put problemsof the week up on the board and wewould try to figure them out all week.she wouldn't help us untill freiday. ithought it was fun.

Yes it gives them a chance to useknowledge that they have tounderstand new problems. i willalways go back and explain it but ithink it is good for kids to use theirbrain to try to figure thinks out on theirown.

once a week, maybetwice it is isn't toochallenging.

1 After providing a strong rationale forletting children solve problems on theirown, this respondent stated that shewould “go back and explain,” indicatingthat she will provide a large part of thereasoning instead of letting the childrendo as much of the thinking as possible.

4 (Scored re Rubric Detail 1C)q5.0 q5.1 q5.1_explain q5.2 Score Comment

I hated it because I like to know whatI was doing, and then I was lost andhad no clue, and would play arounduntil she showed us what to do.

Yes I am interested in how the view theproblem, because sometimes it isbenificial to know where to start from.

Once or twice amonth

1 This respondent indicates that she willprovide students with problem-solvingopportunities for assessment purposesbut does not indicate that she plans touse children’s thinking in herinstruction.

B7–S5

5 (Scored re Rubric Detail 1D)q5.0 q5.1 q5.1_explain q5.2 Score Comment

I can't say I really remember manyteachers giving us new problems andhaving us solve them without theexplanations first. But I do rememberthinking that there must be a trick to it, ora fairly easy way to solve it. I always likedhaving new problems though, because theywere different than the same oldalgorithms, etc. I liked being challenged,but only if the teacher didnt' say" This oneis going to be tough, I'm not sure if any ofyou will figure it out before I explain it..."

Yes I think that this is highly benificial.Some students may get discouraged,but if there was the idea that it'sokay not to understand something atfirst, I think this would not be anissue. Having children think about aproblems tends to spark interest inactually finding out how to do aproblem.

I think it'ssomething youcould actually doevery time youintrodue a newkind of problem.

1 Use the practice to spark interest in theproblem before the students are taughthow to solve it. The rationale for thepractice is to motivate students, not togive them opportunities to think forthemselves as an integral part ofinstruction.

6 (Scored re Rubric Detail 2A)q5.0 q5.

1q5.1_explain q5.2 Score Comment

I was the kind of kid who liked to do things the"right" way, so I always used the traditionalmethods that were taught by my teacher. Mymom was a teacher, too, and she'd teachthings differently (although they got me to thesame answer), but I always resisted herapproach because it wasn't the way my teacherdid it, and I was scared to get it wrong.

Yes I don't want them to be like I was--to thinkthere was only one way to do things. Byallowing them to expand and imaginesolutions, I feel they'll be more open todifferent ways to problem solve, and willend up with a better grasp of the materialand th

Probably onceon each newsubject matter,such as fordecimals,addition,subtraction,etc.

2 Strong rationale for importance ofproviding children with problem-solving opportunities, but thispractice will be employedinfrequently.

7 (Scored re Rubric Detail 2B)q5.0 q5.1 q5.1_explain q5.2 Score Comment

Most likely I would be frustrated andconfused, but I would give a lot of effort intrying to solve the problem.

Yes It is important that children excersise theremind and come up with there own way tosolve a problem. Perhaps create newalgorithms for themselves instead of thestandard ones that I learned in school.

Depending onthe students,maybe oncebefore we start anew concept inmath.

2 Children will devise their ownways to solve problems; theseapproaches will be legitimate.How often this practice will beimplemented is unclear; Response5.2 is tentative.

8q5.0 q5.1 q5.1_explain q5.2 Score Comment

I struggled greatly. It wasalways difficult for me to think ofmy own algorithims for solvingproblems, I always relied on theteacher to show me the steps.

Yes I think this is crucial to the develpoment of childrensmathematical thinking. It encourages them to think of andfind methods that they understand, and it also give themthe oportunity to share their methods with otherclassmates who may be having difficulty seeing how theproblem is solved.

I would askthem to dothid fairlyoften.

3 Respondent provides a strongrationale for having studentsdevise their own strategies.Respondent seems clearlycommitted to this form of practiceand plans to use it consistently.

B7–S5


Aq5.0 q5.1 q5.1_explain q5.2 Score Comment

I was always excited because I love mathand it was easy to me. I would love tojust do math all day. In the 7th and 8thgrade I had a teacher that gave about 10of us a pre-algebra book and told use togo and learn while he taught those whodid not do well in math. I also use tohave so old college math books I wouldtry to solve problems from. Math did notscare me so i welcomed the challenge.

Yes I would ask them to see who could answerit and who may need some assistance. Iwould probably not ask them to share outloud so I would not make a child feel badfor not understanding.

not very often butmaybe

occassionally. I dothis now with some

of my 2nd graders atwork and I

encourage them towork together on

solving the problems.

Bq5.0 q5.1 q5.1_explain q5.2 Score Comment

I would try to solve it but I would usuallyget very frustrated. I did not like notbeing shown how to solve it, but I alwaysremember the solution better if I did it onmy own.

Yes I think that it is important for students tobelieve that they can solve any mathproblem without instruction. i also thinkthat if students solve the problem on theirown, they are more likely to understand itand figure out which solution works bestfor them.

Every time a newconcept is taught.

Cq5.0 q5.1 q5.1_explain q5.2 Score Comment

I would be really surpised and reluctant tous it initially

no I would rather ecplain to them first not asked

Dq5.0 q5.1 q5.1_explain q5.2 Score Comment

i felt that it was a challenge butsometimes got frustrated.

Yes because then they can explore their ownpossibilities bfore i show them a coupleways to do it.

every section

B7–S5

Eq5.0 q5.1 q5.1_explain q5.2 Score Comment

Frustration because I didn’t have thebasis from which to solve the problem.However I also remember these instancesas being the ones from which I learnedthe most. It was a confidence boosterknowing that I had figured out a means tosolve a problem I had no idea about.

Yes The experience of having to reason andfigure out a problem on one’s own oftentimes proves to be beneficial. What wouldbe important in this strategy is to review thepossible solutions that students came upwith and talk about why or why not, this wasa productive or proper means of solving theproblem.

Probably not terriblyoften, because Iwouldn’t want tofrustratestudents…butoftenenougth that theycould see their ownability.

Fq5.0 q5.1 q5.1_explain q5.2 Score Comment

I was scared and nervous. I usually gotthe problem wrong if I attempted it at all.

No I think not showing them how to it will scarethem and maybe in the long run scare themaway from math.

Not asked

Gq5.0 q5.1 q5.1_explain q5.2 Score Comment

Panic! Yes If this was done on a regular basis, thechildren would not panic as I used to do.Also I would say, “who has a different wayof working the problem?” to encouragedivergent thinking and understanding.

Most of the time

Hq5.0 q5.1 q5.1_explain q5.2 Score Comment

Scary – I was able to learn from theteacher’s examples but solving by usingmy own method would be difficult.

Yes I would like to see how and why mystudents used the methods that they choseto use. I believe they would learn easierand get a better understanding by doingthis.

Not too often, butjust enough toenable them toabout it on theirown.

B7–S5


Iq5.0 q5.1 q5.1_explain q5.2 Score Comment

Most of the time it was a challenge waitingto be broken. I always thought it was funto try and work out a problem withoutguidance.

Yes Most problems can be solved in more thanone way. That's the kind of problem I wouldask them to do. I also think it's a good ideafor students to be able to attack a problemusing the skills that they possess. This willallow them to come up with their own solutionand to see that others may have the sameanswer but through a different method.

Almost every day

Jq5.0 q5.1 q5.1_explain q5.2 Score Comment

It was fine you just had to think about it, Ididn't have a problem with a newapproach, it just longer to figure out.

no Some students may get frustrated and thenconfused and it would be a big mess.

Not asked

Kq5.0 q5.1 q5.1_explain q5.2 Score Comment

I feel that it is a new challenge and musttry to do it.

Yes i would ask them why becasue i think its agood idea to have children think hard and letthe m think thier own way and expand theirthinking on math skills. A child can solveproblems on thier ow nway, even withoutshowing them how first.

once in a while, orevry new chapter

Lq5.0 q5.1 q5.1_explain q5.2 Score Comment

I disliked it because I wanted to do it theright way not the wrong way. Unless itwas some sort of game or puzzle.

Yes This would make them use their ownknowledge to create a way to solve it. Theymay do it correctly or discover a newinventive way. Then, if they do it incorrectly,I will explain why and it will be a goodlearning experience and they will improve.

Once in a while,not often, becauseit way bediscouraging forsome.

B7–S5

Mq5.0 q5.1 q5.1_explain q5.2 Score Comment

excitement, anxiety, frustration,accomplishment

Yes If you let students stretch for their ownsolution or method, then they will getpractice as active learners and becomebetter problem solvers. Self Discoverysticks with you a lot longer that being toldprocedures and formulas.

every day

Nq5.0 q5.1 q5.1_explain q5.2 Score Comment

I was confused and did not understandhow to solve the problem,

Yes I would do this because it would give them achallenge. Later on after they have tried, Iwould show them hoe to solve it.

not very often

Oq5.0 q5.1 q5.1_explain q5.2 Score Comment

I usually would either not attempt theproblem and wait for the teacher to showme how to do it or I would try and drawfrom skills I had already learned to solvethe problem.

Yes It is good for them to try and solve a problemby making sense of it in their own head. itgives them a chance to really think about itinstead of following a pattern

sometimes, notevery time though.some things arebetter to beexplained a littlebeforehand sothey are notdiscouraged by itor think it isimpossiblebecause theycouldn't get it ontheir own

Pq5.0 q5.1 q5.1_explain q5.2 Score Comment

unsure if the method was used correctlyor if I would have problems trying to figureout the mehod

no I would like to give them some sort offoundation or background to help them solvea problem. Also I would like to shareexamples of how to solve problems so theycan follow those examples

Not asked

B7–S5


Zq5.0 q5.1 q5.1_explain q5.2 Score Comment

My first reaction is confused because I'mnot sure of how to do the problembecause i have never seen it before.

Yes I would ask them to solve a new type ofproblem in their own way first because it allowsthem to be creative and to find the way theyunderstand. It will also teach them that thereare more ways than one to solve a problem andthat different does not mean wrong.

I think beforeeach newlesson.

Yq5.0 q5.1 q5.1_explain q5.2 Score Comment

I became nervous and didn;t have faith inmyself that I would be able to solve it.

Yes I think that maybe as a warm up it would beinteresting to see if the children could figure itout on their own frist from the knowledge theyalready have.

one every twoweeks

Xq5.0 q5.1 q5.1_explain q5.2 Score Comment

I remember being both frustrated andexcited. Initially it was hard tounderstand the method being proposed,but once I was able to make sense of thenew method, it was exciting to be able tosolve problems in different ways.

Yes I think allowing a student the opportunity todevelop their own understanding for a methodis an important tool. Also, allowing students tostruggle with a concept before having acomplete understanding often leads toexploration and understanding of underlyingideas that may be missed in simply explaininga method.

Often...probablywith introductionof each newconcept or unit.

Wq5.0 q5.1 q5.1_explain q5.2 Score Comment

well i don't remember what my reactionswere when i was younger, but i knowbefore this semister i was reluctant tosolve problems. i did it to get credit but ididn't want to do it. now i am willing to doit because i know there is more than oneway of solving a problem and i might findthat this new was is faster and or easierto compute

Yes well, i want my children to realize what i realizenow and that is there are more ways to solveone problem. and that i might give them a newway that is easier for them to use andunderstand.

as often as i can

B7–S5

Vq5.0 q5.1 q5.1_explain q5.2 Score Comment

I was scared, and I usually did notattempt it until they showed me how.

no I answered wrong, I meant to say yes, justbecasue it scared me does not mean it wouldscare my students. I have a fear of math forother reasons, so I am biased. I would like tothink that in my class my student would not beafraid to try problems and from my recentexperience, I think most children are willing to trythe problems if you ask them to.

not asked

Uq5.0 q5.1 q5.1_explain q5.2 Score Comment

I would figure out my own way ofsolving the problem and if I couldn’tthen I would have to ask the teacherto explain.

No Without knowing how to solve the problem thechild is thrown in to the deep end of the poolwithout a life jacket. Not teaching them a child orany adult how to swim before throwing them in thepool will lead to them drowning and maybe theycould tach themselves to swim but in many casesthey will drift to the bottom of the pool.

Not asked

Tq5.0 q5.1 q5.1_explain q5.2 Score Comment

i became frustrated until the teacherexplained it to me.

Yes i would ask my students to solve a new kind ofproblem so see what kind of algorithms and waysof thinking they come up with.

once in awhile

Sq5.0 q5.1 q5.1_explain q5.2 Score Comment

I actually got pretty frustrated. I didn'tlike not knowing how to do something,to me, fighuring something like this outwas very difficult. If someoneexplained it to me i would most likelyunderstand but if i had to solve it onmy own without knowing how to do it, iwould get frustrated and think myanswer was always wrong.

Yes Although i wasn't fond of the exercise, i think itwould help, especially people who felt like i didwhen i was younger. Maybe they'll understand itand learn how to solve problems on their own. Iwish i could be like that i just wasn't very good atmath.

Only everyonce in a while,i don't want toover do it.

B7–S5



A 1Asking students to solve problems on their own will not be an integral part of instruction but will befor assessment purposes, with little emphasis placed on it.

B 3Strong rationale for asking children to think for themselves. This practice will be an integral part ofinstruction; commitment to this approach is indicated.

C 0 Never ask students to solve problems without first showing them how to solve the problems.

D 1Ask students to solve problems and then show them how to do the problem. Thus children’ssolutions will not be an integral part of instruction.

E 2Strong rationale for having students think for themselves, but indication is that they will beexpected to do so infrequently.

F 0 Never ask students to solve problems without first showing them how.

G 3Brief but clear indication of a strong rationale for having children think for themselves andcommitment to this form of practice.

B7–S5


H 2Having students solve problems on their own will promote understanding and make learningeasier. Strong rationale given for this instructional practice, but it will be used infrequently (thusthe score of 2).

I 3 Convincing evidence that students’ approaches will be used in instruction.

J 0 Convincing evidence that students will not be asked to solve problems for themselves.

K 2Strong rationale for giving students problem-solving opportunities, but they will be giveninfrequently.

L 1Indication that after students explore their ways, the teacher will show them the correct way, so willnot depend on this practice for instructional purposes.

M 3 Strong rationale for problem solving, which will be an integral part of instruction.

N 1Students will be given a chance to think about problems (infrequently), but then the teacher willshow them how to solve them.

O 2 For students to think and make sense of things is important but will not be done often.

B7–S5


P 0 Children will not be given opportunities to independently devise solutions.

Z 3 Strong rationale for having students solve problems on their own and commitment to this practice.

Y 1 Useful to see whether children can “figure out” problems. We interpreted this as assessment.

X 3Important for children to construct their own understanding; strong rationale for this instructionalpractice.

W 2

Although children should learn that there are several ways to solve a problem and children’s wayswill be as legitimate as the teacher’s, the teacher’s ways will also be a focal point of instruction.Thus the response is not scored 3. Because this response does not fit the rubric well, we scoredit holistically. A good reason is provided for giving children opportunities to think for themselves,but the respondent states that ultimately children should see that the conventional approach isbest (a Jekyll-and-Hyde kind of response).

V 1Students should not be afraid. Whether students’ approaches will be an integral part of instructionis unclear. This was a vague response and difficult to score.

U 0 Classic 0—never ask children to solve problems on their own. It may harm them.

B7–S5


T 1 Ask children to solve new kinds of problems for assessment purposes only.

S 2Having children think for themselves will promote understanding and will be an integral, thoughinfrequent, part of instruction.

Scoren % n %

0 56 35% 25 16%1 84 53% 83 52%2 16 10% 30 19%3 3 2% 21 13%Total 159 159

Pre Post


B7-S7


Belief 7

During interactions related to the learning of mathematics, the teacher should allow the children to do as much of the thinking as possible.


The focus in this rubric is on whether the respondent finds the teacher’s guidance to be excessive.She did not provide the child with an opportunity to devise a solution on his own. Instead she told him what to dofor each step. Respondents who are considered to strongly hold the belief that teachers should allow children todo as much of the thinking as possible note in their initial reaction to the clip that the teacher should have giventhe child an opportunity to solve the problem for himself. Others express this view only after being asked todiscuss the weaknesses in the teaching. Some respondents note that the teacher’s guidance is good and thennote that it was excessive. These contradictory responses are interpreted as weak evidence of the belief. Somerespondents find no weaknesses in the teaching at all, and some think that the teacher should do even more tohelp the child.

The difficulty in coding this rubric is in deciding between scores of 1 and 2. Determining whether the respondentis contradicting herself in her responses is sometimes difficult.

B7-S7

Rubric Scores

0. Responses scored 0 include overall satisfaction with the guidance provided by the teacher or suggestions for how theteacher could have helped the child more. These responses are interpreted as showing no evidence for this belief,because the respondent presents no critique of the teacher’s taking over the thinking of the child.

1. Two types of responses are scored 1. One type includes a seeming contradiction. The respondent expressessatisfaction with the guidance provided by the teacher but later suggests that the teacher could have provided less help;thus the respondent indicates some interest in children’s thinking for themselves without being troubled when the teacherdoes not allow for such thinking. The second response type includes a conclusion that the problem presented was toodifficult for the child. The respondent does not address the extent to which the child should be allowed to think for himself,because she thinks that for this problem the child is unable to think for himself.

2. Responses scored 2 do not show satisfaction with the guidance provided by the teacher although the response mightinclude positive comments about the use of manipulatives or the real-world context of the problem. Although therespondent does not comment positively about the help provided by the teacher, she critiques the teacher for leading thechild excessively only when asked to point out the weaknesses in the teaching. She does not raise this concern beforebeing asked about the teaching weaknesses. We interpret this response as weaker evidence for the belief than a critiqueof the teacher’s excessive help in Response 7.1.

3. Responses scored 3 reveal, immediately in Response 7.1, a strong negative reaction to the guidance provided by theteacher. The belief that teachers should allow children to think for themselves is interpreted as being so strong that therespondent notes positive aspects of the episode only when asked to do so in Item 7.2.

B7-S7

Scoring Summary


0A. Overall satisfaction with the guidance provided by the teacher

7.3. No weaknessesB. 7.3. The teacher should do more explaining.

1

A. 7.1, 7.2, or both. Satisfied with teachingTeacher did good job of guiding child (may state that the child was encouraged to think forhimself; seems to like what the teacher did)7.3. Teacher did too much leading or child did not understand7.1 and 7.2 are in contradiction to 7.3.

B. This problem was too hard, AND the child did not understand the concept even after theteacher provided prompting.

2

• 7.1. Teaching is not critiqued for being too leading.7.2. Cubes or story problem or positive reinforcement are strengths; teacher’s guidance notmentioned as strength7.3. Teacher is too leading—child could do more on his own7.1 and 7.2 do not contradict 7.3

3• 7.1. Teacher was too leading.

7.3. Teaching was too leading.Because of the leading, child did not understand.

B7-S7

Examples

1 (Scored re Rubric Detail 0A)q7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

well i think that at first it's only naturalfor kids to get overwhelmed when theyhear a pboblem like this. but with thehelp of manipulatives and theinterviewer he was ble to solve thepbolem . becuase she guided him. be

well, without directly doing theproblem for him. the teacherwas able to help him figure itout for himself. she was alsovert positive and cheerful,which I think hepls.

none 0 Overall satisfaction withthe teaching in the episodeexpressed.

2 (Scored re Rubric Detail 0B)q7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

I think that the child's first answerstood out to me. depending on thechild's grade, i would have thought thathe would have tried to solve theproblem out and get to the correctanswer. but, since he was young, theway he used the blocks was a goodway to solve the problem.

I think that the blocks is agood way of teaching in thisepisode. it gives the child avisual or something to use forcounting rather than trying tosolve it with a pencil andpaper.

Instead of saying "groups" in thebeginning, i would ask the child topretend that you want to put 4 people ineach "car." Because it seems like thechild didnt know why he was taking 4blocks away each time, so that was aweakness. I would maybe ask the childto draw out at least 7 cars on the paperand then put 4 in each car until thereare no more blocks (poeple) left. thentell him to count how many cars heused out of the seven. i think that ifyou give a child a visual or some sortof picture, it would be easier to solveand understand the problem.

0 The teacher should providemore guidance.

3 (Scored re Rubric Detail 1A)q7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

He couldn't solve the problem with outsome scaffolding and somemanipulatives but he did it when he gotwhat he needed.

Use of manipulatives, talkingthrough it with the student,led him where he needed togo.

Maybe led him a little too much be stilla very good job.

1 Response seems to becontradictory. The childcould not have solved theproblem without the helpthat the teacher provided(7.1), but the child wasgiven too much help (7.3).

B7-S7

4 (Scored re Rubric Detail 1B)q7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

i think the problem was too hard for him.he solved it but with a lot of prompting.it didn't look like he really understoodthe problem or what to do and didn't atall understand the concept of division

strengths were that she saidgood job!

the problem is just too hard for him.she also prompted him a lot! he didn'tunderstand the meaning of divisioneither and she didn't really explain it

1 The problem was too hardfor the child.

5q7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

The child just guessed when he wasfirst asked the problem. He didnt'attempt to write it down or solve for theanswer. Then when he used blocks hewas able to understand that 5 carswould be needed for 20 kids.

The strengths were the visualaid, the blocks. The blocksreally help children see theproblem rather then just onpaper.

The teacher should have firtst given theblocks to the child and then told himtheses are 20 kids if four can fit onlyone car how many cars are needed.Then let him figure it out on his own.After he did his understanding thenshow him how to do it if he got it wrong.

2 According to 7.3 the teachershould have let the child tryto solve the problem on hisown, but no critique of theleading given in 7.1. Notscored 1 becausesatisfaction with theteacher’s coaching notexpressed.

6q7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

seemed like the teacher was leadinghim more than he was figuring it out forhimself.

using manipulatives to act outthe word problem

too leading 3 Throughout the response,the teacher is criticized fordoing too much leading.This response indicates abelief in the importance ofallowing children to findsolutions for themselves.

B7-S7


Aq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

once the teacher scaffolded thestudent he was able to make someconjunctures on his own

the teacher guided the studnet well.she did not give too much informationand solve the problem. she alos didnot give too little information so thatthe problem was difficult to do

i think the studnet could havecompleted more of the problem on hisown

Bq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

As I said before manipulatives arevery powerful. They make theabstract concrete.

The teacher used manipulatives andexplained every step of the procedureas she thought out load about theproblem.

No obvious weaknesses.

Cq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

before the child used manipulatives, hereally couldn't even fathom what it was heneeded to do. as soon as themanipulatives were introduced, he seemedto get more of a focus, and with theteacher's help, he got the problem right.

manipulatives made theconcept concrete. the childcould actually see the 4"kids" in each of the 5"cars."

it seemed the teacher led the child to theanswer, and I would want to see if thechild, once introduced to themanipulatives, could have figured outwhat to do with some questioning from theteacher, rather than the teacher saying"now do this, and this etc"

Dq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

Well first the child guessed when theycouldnt think of it on the top of their head.Then he was asked to use blocks. It sortof seemed like he was really being pushedin the direction the interviewer wanted himto go instead of going their on his own.

The teaching had a lot ofpositive reinforcement, Ihave to give it that. Shewas very positive when hedid something and sheseemed really positivethroughout the entire video.

She pushed the child a little too hardinstead of letting him go there on his own.Maybe it was because he wasn't going tofind it on his own, but I really dont thinkhe knew why he was counting all thegroups of four, other than because hewas told to.

B7-S7

Eq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

I'm not sure that the child actuallyunderstod what he had done. It seemedas though the interviewer led him throughthe solution and the child was justfollowing the steps. I wonder if he wouldbe able to apply this method to similarproblems.

The teacher's encouragementand ability to stay with thechild seemed a strength.

Again, I'm not sure that the way theinterviewer led the child through thesolution really added to hisunderstanding. He was able to derivethe correct answer, but I don'tnecessarily think that was the mostimportant part of the interview.

Fq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

The child worked the problem out withblocks and got the right answer. He did agood job.

The blocks helped him to seewhat the problem was allabout.

The teacher spoon fed him theapproach. She could have backed off tosee what he could do with less help. Hedidn’t get a chance to think for himself.

Gq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

Yes, children need math manipulatives,especially at a young age when they can'tthink abstractly. It was also good on theteacher's part to mention to the child hedid an "adult" math problem. It made thechild feel smart and good about math.

She complimented the child.She was kind, encouraging, andshe used manipulatives to helphim solve the math problem.She was also very pacient.

I didn't see any.

Hq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

What stood out for me was how mucheasier it was for the chidren to solve theproblem when they had the cars tomanipulate and count for a total.

Allowing the children to manipulateand asking good questions to helpthem think about the problem andlead them along the right path.

Sometimes the teacher seemed tobe leading them too much to theanswer and not allowing them tothink on their own.

B7-S7

Iq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

The teacher was very encouraging.Even when the child guessed 10cars, she applauded his efforts.Upon seeing the groups of blocksdivided up as "cars" he understoodthe concept. Visualization is veryimportant for later on when he reallybegins to perform divisionproblems.

The instructor first related the word problem tohim, making the child feel like they areplanning a camping trip for his classroom andthey need enough cars to accomodate hisclassmates. She encouraged him and gavehim praise, such as "you're such a goodcounter." She identified when he was guessingbut did not give him a hard time about it,instead she persisted till he understood andmade him feel like he had performed "big kidmath." That's good for his confidence which isimportant in perfoming math, especially infront of someone, so it should help him in hisclassroom.

Perhaps she should havestepped back a little and let himexplain how he intended tomake the 20 blocks into 5 cars,he may have been able to do itwithout her touching the blocksand sorting them around forhim.. After there were 20blocks pulled out I would haveliked to see what he did nextwithout her prompts.

Jq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Commentwhat stood out what how thestudent didnt try to come up with asolution at the beginning he justguessed. that is what moststudents would do at his age. also,i liked the way the teacher helpedhim through the problem but i don'tthink he understands and would beable to do another problem on hisown.

The teacher did a good job complimenting thestudent and helping him through the motions.

the teacher could haveexplained that you have to splitup the kids into each car byfours and then told him to putthe blocks in groups. maybe ifhe did more of it on his owninstead of her telling himdirectly each step, he wouldhave learned it more.

Kq7.1 q_video_7.2_reaction q_video_7.3_reaction Score CommentThe manuplatives worked great inthis problem. When te child wasasked to solve the problem, heguessed. With a little coaxing itwas easy for the child to see thatfour kids fit in each car.

the interviewer coaxed him, by aking him toshow her the 20 cars. then asked to seegroups of 4. he even stated that the group offour was easy. She didnt tell him the answerbut helped him to understand what he wasgrouping and he was able to see and solvethe problem

I don’t think there was any

B7-S7

Lq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

Well, I found myself curious toknow how that kid would havesolved the problem if he had beengiven a chance. The method theteacher walked him through wasone that many kids probably do.But I saw it coming from theteacher and not the kid. It wouldhave been interesting to see whathe would have done if she hadprobed his first answer (10 cars) abit more, perhaps leading to therealization that 10 cars were toomany. What would he have donethen?

It was great that she asked him how hefigured out his first answer of 10 cars. Askinghim to show the 20 kids was also potentiallyhelpful. In addition, it was nice that she didnot reveal by her demeanor whether hisanswer was correct.

She led him through a methodof her choice. This methodseems like a great one, but hemight have been able to figurethis problem out himself. Shecould have asked questionsabout his initial answer,possibly leading him to aperturbation, which might haveresulted in him thinking aboutthe problem differently or findinghis own solution. Her behaviordid not allow him thatopportunity.

Mq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

It was obvious that the student wastaking a complete guess at theanswer and did not understand thedivision question. I was surprisedthat he did not attempt to use theblocks at first or work the problemout with the paper and pencil.

The teacher was friendly, she tried to makethe question fun, and she made solving thequestion fun.

I think the teacher should haveprompted the student to utilizethe materials on the table ifnecessary. I think it ispossible that the student couldhave solved the problemwithout help if he were told touse the blocks.

Nq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

The interviewer prompted the childa lot. On his own I don't think hecould have solved it, but it wasgood that she had him think itthrough. With the prompts he couldfigure it out.

The interviewer did not give him the answer,she just prompted him in the right direction.That was good.

She could have given him alittle more time to think abouteach step himself beforeprompting him.

B7-S7


Zq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

The teacher asks the student how hefigured it out,and then she had thestudent work it out with blocks so hecould see what he was doing.

She gets the child to correct himself byasking leading questions that make thechild think

The teacher really led thestudent, and didn't let him try tocorrect himself on his own

Yq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

I thought it was good that she let himtry and answer the problem first andthen she showed him how to figure itout using the blocks. I think this is avery important part of a childslearning. They need to test thingsout themselves and then see thedifferent ways to approach a problem.

I think the strengths of this video wereallowing the child to think on his own andsolve the problem. She helped see how tofigure it out without just telling him hisanswer was wrong and telling him thecorrect one.

I didn't see any weaknesses inthis video clip. I really liked it.

Xq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

at first the boy guessed the answer,but with the use of manipulatives hewas able to solve the problem.manipulatives are a useful tool inhelping children to understand mathproblems. with help from theinstructor, the student was able tounderstand the division problem. oftentimes, visual aid works great

there were a couple of strengthens to thisteachers approach in helping the boysolve the division problem. sheconstantly gave postitive feedback to theboy which was great, not only did shegive postitve reinforcement, but alsohelped the boy in showing himmanipulatives. and how to use them.she also took her time, and did not rushthe boy into solving an answer quickly.

there were a couple ofweaknesses by the teacher notwriting the probelm down onpaper. she told the boy theprobelm out loud. seeing howthe child is a visual learner, itwould have been to his benefit tosee the problem, instead ofvisualtaion. while the use ofmanipulatives was great, shejumped in and showed the boythe manipulatives and how to usethem. she shoudl of waited a bitlonger to see how the boy wasging to solve the problem,instead, of rushingh right to thecubes

B7-S7

Wq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

When the interviewer first aske dthequestion the child guessed. This tome shows that he doesn't understandhow to solve the problem. I don't thinkhe understood even after they countedout 20 blocks and seperated them into5 piles.

I think it was good to suggest that hecount out 20 kids. I think a student mightneed this liitle push to get them started,and he may need a few more triggerquestions i.e. Can you show me 4 kids inone car?

I think this particular episodewas very guided. It appearedthat the child came up with theanswer because he followed whatthe interviewer said but I don'tthink he understands what hedid.

Vq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

nothing reallt stood out to me. itseems like the instructor helped outthe student a lot

good tone of voice and lots ofencouragement

too much help for the student

Uq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

I noticed that at first the boy justguessed but then he was able toanswer the problem using cubes andhaving assistance from the teacher. Ithink it as a good idea for the teacherto help him out and introduce theblocks. It gave the child something tovisualize when answering.

The teacher gave good positivereinforcements. She also encouraged thechild to think about his answer and whathe was thinking. She helped him out justenough so that she was not answeringthe problem for him but assisting him insolving it himself. The teacher brokedown the problem with the blocks andmade it easy for the child.

I did not really see too manyweaknesses in this clip. Maybethe teacher could have found outmore about why the child thoughtthe answer was 10 at first.

Tq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

He understood the division problemwith the help of manipulatives.Division is an area that may not becovered in his grade but he was ableto do it. I was impressed with thechild.

The teacher was clear in what she wantedfor the problem. She used manipulativeswhich helped the child see the problemand be able to figure it out.

The teacher may have helped toomuch. She could have waited afew minutes and seen if the childcould figure out the problem ontheir own. Children are smarterthan we give them credit for.

B7-S7

Sq7.1 q_video_7.2_reaction q_video_7.3_reaction Score Comment

i liked that he was getting excited whenhe was solving the problem.. also likedthat the teacher encouraged himthroughout each step, this really helpsa child fell good and want to continue, ifeind it funny that a child always seemsto take a guess first before theyactually solve

throughout the whole clip she wasencouraging him and tell him good job.. ifeel this is very important.. i liked thatshe started to count but hten let thechild finish and come to the answer

i didn't like that she said "This iswhat the big kids do" thecomment could go two ways.. itcould encourage him, but alsomake him scared

B7-S7



A 1Response 7.2 indicates satisfaction with the guidance provided by the teacher; but 7.3 indicatesthat perhaps too much guidance was given (called a Jekyll and Hyde response).

B 0 Complete satisfaction with the guidance offered by the teacher indicated.

C 2Neither satisfaction nor dissatisfaction with the guidance provided by the teacher is expressed in7.1 and 7.2. Comments on strengths limited to satisfaction with use of manipulatives. Critique in7.3 is consistent with other comments about the video.

D 3Concerns expressed in 7.1 (before question on weaknesses was posed) about the guidance ofthe teacher. The teacher guidance was noted right away and elaborated on in 7.3.

E 3That the teacher’s guidance did not build the child’s understanding is noted in the initialresponse. View of the importance of teachers’ giving children opportunities to devise solutionsfor themselves is sufficiently strong to be of concern immediately.

F 2Satisfaction with the guidance offered by the teacher is not expressed at all, but that the teachershould have let the child devise a solution for himself is not stated until 7.3. Responses notcontradictory, but the dissatisfaction with the teacher is only in 7.3.

G 0 Satisfaction with the guidance offered by the teacher is expressed.

B7-S7


H 1Satisfaction with the guidance offered by the teacher expressed in 7.2. Concerns about theguidance noted only in 7.3.

I 2Positive comments in 7.2 are about using a relevant context and providing positive reinforcement,not specifically about the guidance provided. Concerns about excessive guidance mentioned in7.3.

J 1 Level of guidance praised in 7.1 and 7.2 and then criticized in 7.3.

K 0 Only praise is expressed for the teaching.

L 3That the teacher took over the thinking of the child instead of letting him try to solve the problemon his own is expressed in 7.1.

M 2That the teacher might have offered less help is mentioned in 7.3, but the guidance is neithercriticized not praised in 7.1 and 7.2.

N 1In 7.1 and 7.2 satisfaction with the guidance offered by the teacher is expressed. In contradiction,in 7.3, the respondent suggests that the teacher could have waited before offering herassistance.

Z 1

Inconsistent statements are made about the guidance offered by the teacher—teacher askedleading questions that made the child think (7.2) but the teacher really led the student (7.3).Whether the respondent is satisfied with the teacher’s guidance is unclear, so the responses isscored 1.

B7-S7


Y 0Respondent values children’s opportunities to devise solutions but states that the child in thevideo was allowed to think on his own to solve the problem. She expresses no concern about theguidance offered by the teacher.

X 1Manipulatives are the focus of the response to 7.1. Approval of the teacher’s guidance expressedin 7.2: The teacher took her time and did not rush the boy. In 7.3 respondent contradicts thisstatement and suggests that the teacher not rush the boy to use the manipulatives.

W 2

Although the respondent states that the child needed a little push to get him started and a fewtrigger question, she criticizes the teaching as being too guided. Thus, she is dissatisfied withthe teacher’s guidance and notes that the child does not understand his work. Mention of herconcern about the guidance offered by the teacher in 7.1 would be interpreted as strongerevidence of this belief. This ambiguous response could be scored 1, but we interpreted thecomments in 7.2 to mean that the respondent favored asking only a few trigger questions andconsidered the number of guiding questions in this interaction excessive.

V 3This limited response indicates, in 7.1 and 7.3, criticism of the guidance offered by the teacher.We scored this response a weak 3.

U 0The suggestion that the teacher might have pursued the child’s guess is insufficient evidence ofthe belief to score the response 1. The respondent states (7.2) that the teacher gave just the rightamount of help. Indication that the teacher helped too much is required to score 1.

T 2

That the teacher may have helped too much is noted in 7.3, but in 7.1 and 7.2, dissatisfaction withthe teacher’s help is not expressed. We interpreted the ambiguous statement “'The teacher wasclear in what she wanted for the problem” to mean that the teacher was prepared withmanipulatives for the child to use.

S 0 No concern is expressed about the guidance offered by the teacher.

Scoren % n %

0 106 67% 77 48%1 26 16% 45 28%2 26 16% 24 15%3 1 1% 13 8%Total 159 159

Pre Post


IVMinimum Software

Requirements

Minimum System Requirements for Administrator and Participants

After the IMAP Web-Based Beliefs Survey has been installed on a web server by a network administrator, it is designed tobe accessed on the Internet from any PC or Macintosh equipped with the following minimum software requirements:

Apple Macintosh

Macintosh OS 8.1 or laterMicrosoft Internet Explorer 5 or laterApple QuickTime 4 or laterMicrosoft Excel 98 or later (Survey Administrator Only)

PC

Microsoft Windows 95 or laterMicrosoft Internet Explorer 5 or laterMicrosoft Windows Media Player 6 or laterMicrosoft Excel 98 or later (Survey Administrator Only)

(Note that a separate list of software requirements for installation is provided on the installation disk.)

VInstallation

Installation

The IMAP Web-Based Beliefs Survey is designed to be installed on a web server and should be installed by your networkadministrator or information-technology staff. Separate instructions and technical specifications are include in theDesign Document on the installation disk.

VISurvey Functions

Survey Functions

The Integrating Mathematics and Pedagogy (IMAP) Web-Based Beliefs Survey includes several functions to provide aresearcher flexibility when measuring beliefs. By using the Accessing Data functions, you can view data collection in realtime and easily download collected data. All administrator functions are protected by a password that can be changedand maintained.

By typing admin.asp after the address or URL of the survey (for example, http://www.imapsdsu.org/admin.asp), you canaccess the Survey Administrator screen.

Before you can continue to the Survey Administrator screen, you will be prompted to enter a password. The defaultpassword is password.

Customizing the Survey

We have created three options for customizing the survey:

1. (By belief) From the Beliefs list, select the beliefs of interest to you; the appropriate segments of the instrument will beautomatically assembled for you.

2. (By segment) From the Segments list, select the segments to include.

3. (By both) Select some beliefs and some segments.

After you have selected from the lists of Beliefs and Segments, click Submit at the bottom of the screen to save yoursettings. A message will be displayed to confirm your settings.

Browsing the Survey

To see a table of all the items that comprise the Beliefs Survey, use Browse the Survey, found on the SurveyAdministrator screen. Click on an item to display that section of the survey. A set of navigation buttons is located at thebottom of each page. Click the arrows to move backward or forward in the survey. Between the arrows is a button youmay use to return to the table of items. Responses cannot be submitted from this version of the survey.

Accessing DataAt the bottom of the Survey Administrator screen click Accessing Data for three options for browsing data:

1. Use Monitor Data Collection to see a list of files corresponding to each participant in your study. Simply click on thename of the participant of interest to view his or her responses. Using this function while the person is completing thesurvey will not interfere with his or her work on the survey.

2. The View Data Table function will display, in your web browser, a table of all data collected.

3. The Download Data function will download onto your desktop an Excel spreadsheet named readdatafiles.asp, whichwill contain all collected data. The file will not have the traditional .xcl or .xl file ending, but it will be recognized by Excel.

Password Maintenance

The process of setting and maintaining the password is relatively easy and is similar to the process used in otherpassword-protected applications. To change your admin password, enter the current password and then enter the newpassword twice. Finally, click Submit to save your new password.

Additional Features

Restarting an Incomplete Survey

Participants can restart an incomplete survey, if necessary. This function is triggered automatically when a participantenters his or her name into the intro page exactly as on the previous attempt. The duplicate name will be recognized, andthe participant will be asked whether he or she wants to continue the previously started survey or begin a new survey. Ifmultiple files with the same participant name are found, a list will be provided with corresponding date and times so thatthe participant can select the appropriate survey to complete.

Video Streaming

The IMAP Web-Based Beliefs Survey is designed to work on either PC or Macintosh operating systems, utilizing either afast (DSL, CABLE, or T1) or a slow (56K Modem) connection. The survey is designed to automatically detect thecomputer's operating system and employ the appropriate video-streaming software. Apple QuickTime is used onMacintosh. Windows Media Player is used to stream video on PCs or Windows-based operating systems.

Administration Login

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Password:

SURVEY ADMINISTRATOR

You have three options for cusomizing the survey:

(1) From the Beliefs list, select the beliefs of interest to you; the appropriate segments of the instrument will be automatically assembled for you.(2) From the Segments list, select the segments to include.(3) Select some beliefs and some segments.

BELIEFS

1. Mathematics is a web of interrelated concepts and procedures (school mathematics should be too).

2. One's Knowledge of how to apply mathematical procedures does not necessarily go with understanding the underlying concepts. That is, students or adults may know a procedure they do not understand.

3. Understanding mathematical concepts is more powerful and more generative than remembering mathematical procedures.4. If students learn mathematical concepts before they learn procedures, they are more likely to understand the procedures when they learn them. If they learn the procedures first, they

are less likely ever to learn the concepts.

5. Children can solve problems in novel ways before being taught how to solve such problems. Children in primary grades generally understand more mathematics and have more flexible solution strategies than the teachers, or even their parents, expect.

6. The ways children think about mathematics are generally different from the ways adults would expect them to think about mathematics. For example, real-world contexts support children's initial thinking where symbols do not.

7. During interactions related to the learning of mathematics, the teacher should allow the children to do as much of the thinking as possible.

Beliefs About Learning or Knowing Mathematics, or Both

Beliefs About Children's (Students') Learning and Doing Mathematics

SEGMENTS 2. Missing-addend problem in Pokemon-cards context 7. Video 1 - Teacher working with boy on 20 + 4 measurement-division story problem.

3. Six examples of student strategies to solve 149 + 286. 8. Fraction operations and comparison problems, with and without context.

4. Two children's written work for solving 635 - 482. 9. Video 2 - Teacher shows child fraction-division algorithm. Video 3 - Child attempts to use algorithm 3 days after instruction.

5. When you are a teacher, will you ask your students to solve novel problems? 10-15. Biographical Likert-scale questions.

6. How sure are you that you want to become a teacher? Segments 6 and 10-15 do not assess beliefs. They are included to collect biographical information.

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The IMAP project was funded by a grant from the National Science Foundation (REC-9979902) under the Interagency Education Research Initiative.

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Randy Philipp - last accessed: 1/14/2003 5:58:04 PM

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Web-Based Beliefs Survey

Browsing the IMAP Web-Based Beliefs Survey

In this section of the manual, you may locate any survey item from the Table of Contents or may use the Acrobat arrows to page throughthe entire survey to read the survey items. The version of the survey included in this manual is only for browsing; no data can beentered or submitted. The message “Button not supported in Manual” will appear if you try to submit data.

Table of Contents for Browse VersionThe survey items are listed and briefly described in the Table of Contents. Each item is linked to its item number in the Table of Contentsand can be accessed by clicking on the link. Movement within this version of the survey is possible from the Table of Contents to anypage as well as from page to page (using the Acrobat page-forward or page-back arrows). Bookmarks have been nested under the Tableof Contents bookmark so that you can select any of the three pages of the Table of Contents.

Branching is the one feature of the survey not available for examination in this version. For each item that leads to branching in thesurvey, one response was chosen to be shown in this version. For example, in Item 2, respondents are asked whether a first-grade childcould solve a certain problem. Within the actual survey, the next page the person sees is determined by his or her response to thatquestion. Within this version of the survey, we include only one of the branches (in this case, the one taken if the respondent answers inthe affirmative).

Beliefs SurveyTo page through the entire survey, go directly to the Web-Based Survey. Use the Acrobat arrow at the bottom of each page to advance tothe next page of the survey. Do not enter data or hit the Submit buttons.

Although data cannot be entered in this version of the survey, the video clips on which Items 7 and 9 are based can be viewed. Simplyclick the links to the video to watch the clips. Check the system requirements for accessing the video in the Minimum SystemRequirements. Apple QuickTime 4 or later must be the Internet Explorer preference selected for viewing the QuickTime videos (whichare .mov files) on Macintosh computers. Microsoft Windows Media Player 6 or later must be used to view the videos (which are .wmvfiles) on PC computers.

TABLE OF CONTENTS FOR BROWSING THE SURVEY

Survey Question Content Question Description

Intro page General demographic information

2.1 Missing-addend problem regarding Pokemon cards

Do you think that a first grader could solve this?

2.2 Support your position on the Pokemon question.

2.3 Multiplication problem regarding number of sticks of gum

Do you think that a first grader could solve this?

2.4 Support your position on the gum question.

3.1 Six children's strategies for solving the problem 149 + 286

View all 5 strategies; indicate whether they make sense.

3.2–3.5 Which would you like to see shared?

In what order would you share them?

Comment on how the traditional strategy compares to others.

4.1–4.5 Two children's written work for solving635 – 482

Comparing traditional and nontraditional approaches

4.6–4.10 Further probing to compare the two approaches for subtraction

5.0 What were your reactions when a teacher asked you to solve a new type of problem?

5.1 Question When you are a teacher, will you ask your students to solve novel problems?

5.1 Explanation–5.2 Please elaborate your answer. How often would you have students solve novel problems?

6 How sure are you that you want to become a teacher?

7.1 Video I Did anything in the video clip stand out for you?

7.2–7.3 Comment on strengths and weaknesses of the teaching.

7.4–7.5 Do you think that the child could have solved the problem with less help?

8.1 Fraction operations vs. word problems

Rank the difficulty of four fraction problems, presented in terms of mathematical symbols and also in contexts of story problems.

8.2 Explain your ranking.

8.3 How are you thinking about understanding?

8.4 Explain further what you mean by understand.

9.1–9.3 View video II Write your reactions. Would you expect this student to be able to solve another similar problem?

9.4–9.6 View video III What happened?

10–16 Questions regarding views and opinions on background in and the nature of mathematics

Welcome to the IMAP survey. The purpose of this survey is to help University educators better understand students' orientations to mathematics. Your course grade will in no way be affected by your responses, but we appreciate your efforts to answer each question as best you can.

NOTES:

1. You cannot go back and change answers in this survey so be sure that you have responded to all of the questions before you hit the "submit" button.

2. You MUST use Microsoft Internet Explorer 5.0 or higher. If you do not have this version, click here to download it.

If you are a student, click here to begin ->>>

http://www.microsoft.com/Education/product/ie_benefit.asp

First Name

Last Name

Phone

E-Mail Address

1. Class for which you are completing this survey




Yes

No

3. Teachers often ask children to share their strategies for solving problems with the class. Read the following student answers and indicate whether each makes sense to you. Then, click on the button at the bottom of the page to continue.

Carlos 149 + 286

Written on paper

Henry 149 + 286


Elliott149 + 286

Written on paper

Sarah149 + 286


Does Carlos's reasoning make sense to you?

Yes No

Does Henry's reasoning make sense to you?

Yes No

Does Elliott's reasoning make sense to you?

Yes No

Would you like to see a further explanation?

Yes No

Click here to see a further explanation.

Does Sarah's reasoning make sense to you?

Yes No

MariaManipulatives

= 100Called a flat

= 10Called a long

= 1Called a single




Then I counted like this, '100, 200, 300'; then for the longs, '310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420'; then the singles, '421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435.'


Does Maria's reasoning make sense to you?

Yes No

Carlos149 + 286

Written on paper

Henry149 + 286


Elliott149 + 286

Written on paper

Sarah149 + 286


MariaManipulatives

= 100Called a flat

= 10Called a long

= 1Called a single






Carlos Yes

No

Henry Yes

No

Elliott Yes

No

Sarah Yes

No

Maria Yes

No


330, 340, 350, 360, 370, 380, 390, 400, 410, 420'; then the singles, '421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435.'


First:

Second:

Third:

Fourth:

Fifth:







Ariana

635 – 400 = 235235 – 30 = 205205 – 50 = 155155 – 2 = 153 482



Yes No


Yes No


Lexi Ariana

Why?




Ariana

635 – 400 = 235235 – 30 = 205205 – 50 = 155155 – 2 = 153 482



4.6 Of 10 students, how many do you think would choose Lexi's approach?

of 10 students would choose Lexi's approach.




4.8 Of 10 students, how many do you think would choose Ariana's approach?

of 10 students would choose Ariana's approach.





Yes

No

6. At this point in time, which of the following best captures your position?

I am sure that I want to become a teacher.

I think that I want to become a teacher.

I am not sure that I want to become a teacher.

I am leaning against becoming a teacher.

I am sure that I do not want to become a teacher.


a) Understand


b) Understand



are they same size?


8.3. In a previous question, you were asked to rank the difficulty of understanding .

By understand, were you thinking of the ability to get the right answer?

Yes

No

8.4 On the last question you indicated that you were thinking about understanding as "getting the right answer." Were you also thinking of anything else? Please explain.





Yes No





10) When you were a child, how did you feel about mathematics?

Very comfortable Somewhat comfortable Neither comfortable nor

anxious Somewhat anxious Very anxious

11) When you were a child, how well did you like mathematics?

I loved it I liked it I felt neutral I disliked it I hated it

12) When you were a child, how successful were you in your mathematics classes?

Usually successful More successful than

not Successful about half

the time More unsuccessful than

not Usually unsuccessful

13) Which statement best describes your elementary-school experiences with word problems?

They were very difficult. They were somewhat

difficult. They were neither easy

nor difficult. They were somewhat

easy. They were very easy.

14) Is there something else about your experiences learning mathematics in elementary school that you would like to share?

15) Choose the description that best fits the majority of your mathematics experiences in elementary school.

a) The teacher explained a way to solve problems. The children independently practiced solving several problems. Hands-on materials were never used.

b) The teacher sometimes explained things. Sometimes the children figured out their own ways to solve problems. The children occasionally used hands-on materials. Children occasionally worked in groups.

c) The teacher gave the children problems that they figured out on their own. The children often used hands-on materials and talked to each other about mathematics.

16) Describe the characteristics of someone who is good at doing mathematics.

You have completed the survey.

Thank you very much for your time!

Explanation of Elliott's method

Elliott adds the hundreds together and writes 300.Elliott adds 8 tens and 4 tens and writes 120.Then he adds the 9 ones and 6 ones and gets 15.Then he adds 300 + 120 + 15 and writes 435.

149+ 286

300120

15435

Click here to return to the survey page.

About IMAP

Integrating Mathematics and Pedagogy (IMAP) is a federally funded, 3-year project designed to integrate informationabout children's thinking about mathematics into mathematics content courses for college students intending tobecome elementary school teachers.

Principal InvestigatorsRandolph Philipp, San Diego State UniversityJudith Sowder, San Diego State University

DirectorsRandolph PhilippBonnie Schappelle

Video ProducerCandace Cabral

Copyright © 2003 by the San Diego State University Foundation.

The production of these materials was undertaken as part of a larger project, Integrating Mathematics and Pedagogy (IMAP), in which we studied theeffectiveness of early field experiences in mathematics. The project was supported by a grant from the National Science Foundation (NSF) (REC-9979902). The views expressed are those of the authors and do not necessarily reflect the views of NSF.

IMAP Web-Based Beliefs Survey Manual

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